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== 2.1#h(0.6em)Solve Equations Using the Subtraction and Addition Properties of Equality

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Before you get started, take this readiness quiz.

Evaluate #math.equation(block: false, alt: "x plus 4")[$x + 4$] when #math.equation(block: false, alt: "x equals −3")[$x = −3$]. #linebreak() 
If you missed this problem, review .

#solutionbox[
#math.equation(block: true, alt: "1")[$1$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Evaluate #math.equation(block: false, alt: "15 minus y")[$15 − y$] when #math.equation(block: false, alt: "y equals −5")[$y = −5$]. #linebreak() 
If you missed this problem, review .

#solutionbox[
#math.equation(block: true, alt: "20")[$20$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify #math.equation(block: false, alt: "4 open parenthesis 4 n plus 1 close parenthesis minus 15 n")[$4 ( 4 n + 1 ) − 15 n$]. #linebreak() 
If you missed this problem, review .

#solutionbox[
#math.equation(block: true, alt: "n plus 4")[$n + 4$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Translate into algebra “5 less than #math.equation(block: false, alt: "x")[$x$].” #linebreak() 
If you missed this problem, review .

#solutionbox[
#math.equation(block: true, alt: "x minus 5")[$x − 5$]
]
]


=== Verify a Solution of an Equation

Solving an equation is like discovering the answer to a puzzle. The purpose in solving an equation is to find the value or values of the variable that make each side of the equation the same – so that we end up with a true statement. Any value of the variable that makes the equation true is called a solution to the equation. It is the answer to the puzzle!

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Solution of an equation]

A #strong[solution of an equation] is a value of a variable that makes a true statement when substituted into the equation.
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[To determine whether a number is a solution to an equation.]

+ Substitute the number in for the variable in the equation.
+ Simplify the expressions on both sides of the equation.
+ Determine whether the resulting equation is true (the left side is equal to the right side)
- If it is true, the number is a solution.
- If it is not true, the number is not a solution.
]

#examplebox("Example 1")[][
Determine whether #math.equation(block: false, alt: "x equals the fraction 3 over 2")[$x = frac(3, 2)$] is a solution of #math.equation(block: false, alt: "4 x minus 2 equals 2 x plus 1")[$4 x − 2 = 2 x + 1$].

#solutionbox[
Since a solution to an equation is a value of the variable that makes the equation true, begin by substituting the value of the solution for the variable.

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#figure(figph[A linear equation is displayed, showing 4x minus 2 equals 2x plus 1, ready to be solved for the variable x.], alt: "A linear equation is displayed, showing 4x minus 2 equals 2x plus 1, ready to be solved for the variable x.", caption: none)]),
  [#figure(figph[The image shows the text 'Substitute 3/2 for x.' The fraction 3/2 is written with the numeral 3 in red at the top and the numeral 2 in red at the bottom, separated by a horizontal line.], alt: "The image shows the text 'Substitute 3/2 for x.' The fraction 3/2 is written with the numeral 3 in red at the top and the numeral 2 in red at the bottom, separated by a horizontal line.", caption: none)], [#figure(figph[A mathematical equation asks whether 4 times (3/2) minus 2 is equal to 2 times (3/2) plus 1. The fraction 3/2 is highlighted in red.], alt: "A mathematical equation asks whether 4 times (3/2) minus 2 is equal to 2 times (3/2) plus 1. The fraction 3/2 is highlighted in red.", caption: none)],
  [Multiply.], [#figure(figph[An arithmetic equation where 6 minus 2 is compared to 3 plus 1, with a question mark over the equals sign.], alt: "An arithmetic equation where 6 minus 2 is compared to 3 plus 1, with a question mark over the equals sign.", caption: none)],
  [Subtract.], [#figure(figph[The number 4 equals 4, confirmed with a checkmark, indicating a correct mathematical statement or a balanced equation.], alt: "The number 4 equals 4, confirmed with a checkmark, indicating a correct mathematical statement or a balanced equation.", caption: none)],
))

Since #math.equation(block: false, alt: "x equals the fraction 3 over 2")[$x = frac(3, 2)$] results in a true equation (4 is in fact equal to 4), #math.equation(block: false, alt: "the fraction 3 over 2")[$frac(3, 2)$] is a solution to the equation #math.equation(block: false, alt: "4 x minus 2 equals 2 x plus 1")[$4 x − 2 = 2 x + 1$].
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Is #math.equation(block: false, alt: "y equals the fraction 4 over 3")[$y = frac(4, 3)$] a solution of #math.equation(block: false, alt: "9 y plus 2 equals 6 y plus 3")[$9 y + 2 = 6 y + 3$]?

#solutionbox[
no
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Is #math.equation(block: false, alt: "y equals the fraction 7 over 5")[$y = frac(7, 5)$] a solution of #math.equation(block: false, alt: "5 y plus 3 equals 10 y minus 4")[$5 y + 3 = 10 y − 4$]?

#solutionbox[
yes
]
]


=== Solve Equations Using the Subtraction and Addition Properties of Equality

We are going to use a model to clarify the process of solving an equation. An envelope represents the variable – since its contents are unknown – and each counter represents one. We will set out one envelope and some counters on our workspace, as shown. Both sides of the workspace have the same number of counters, but some counters are “hidden” in the envelope. Can you tell how many counters are in the envelope?

#figure(figph[This image illustrates a workspace divided into two sides. The content of the left side is equal to the content of the right side. On the left side, there are three circular counters and an envelope containing an unknown number of counters. On the right side are eight counters.], alt: "This image illustrates a workspace divided into two sides. The content of the left side is equal to the content of the right side. On the left side, there are three circular counters and an envelope containing an unknown number of counters. On the right side are eight counters.", caption: [The illustration shows a model of an equation with one variable. On the left side of the workspace is an unknown (envelope) and three counters, while on the right side of the workspace are eight counters.])

What are you thinking? What steps are you taking in your mind to figure out how many counters are in the envelope?

Perhaps you are thinking: “I need to remove the 3 counters at the bottom left to get the envelope by itself. The 3 counters on the left can be matched with 3 on the right and so I can take them away from both sides. That leaves five on the right—so there must be 5 counters in the envelope.” See  for an illustration of this process.

#figure(figph[This figure contains two illustrations of workspaces, divided each into two sides. On the left side of the first workspace there are three counters circled in purple and an envelope containing an unknown number of counters. On the right side are eight counters, three of which are also circled in purple. An arrow to the right of the workspace points to the second workspace. On the left side of the second workspace, there is just an envelope. On the right side are five counters. This workspace is identical to the first workspace, except that the three counters circled in purple have been removed from both sides.], alt: "This figure contains two illustrations of workspaces, divided each into two sides. On the left side of the first workspace there are three counters circled in purple and an envelope containing an unknown number of counters. On the right side are eight counters, three of which are also circled in purple. An arrow to the right of the workspace points to the second workspace. On the left side of the second workspace, there is just an envelope. On the right side are five counters. This workspace is identical to the first workspace, except that the three counters circled in purple have been removed from both sides.", caption: [The illustration shows a model for solving an equation with one variable. On both sides of the workspace remove three counters, leaving only the unknown (envelope) and five counters on the right side. The unknown is equal to five counters.])

What algebraic equation would match this situation? In  each side of the workspace represents an expression and the center line takes the place of the equal sign. We will call the contents of the envelope #math.equation(block: false, alt: "x")[$x$].

#figure(figph[This image illustrates a workspace divided into two sides. The content of the left side is equal to the content of the right side. On the left side, there are three circular counters and an envelope containing an unknown number of counters. On the right side are eight counters. Underneath the image is the equation modeled by the counters: x plus 3 equals 8.], alt: "This image illustrates a workspace divided into two sides. The content of the left side is equal to the content of the right side. On the left side, there are three circular counters and an envelope containing an unknown number of counters. On the right side are eight counters. Underneath the image is the equation modeled by the counters: x plus 3 equals 8.", caption: [The illustration shows a model for the equation #math.equation(block: false, alt: "x plus 3 equals 8")[$x + 3 = 8$].])

Let’s write algebraically the steps we took to discover how many counters were in the envelope:

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#figure(figph[A simple algebraic equation: x + 3 = 8.], alt: "A simple algebraic equation: x + 3 = 8.", caption: none)]),
  [First, we took away three from each side.], [#figure(figph[The equation x + 3 - 3 = 8 - 3, demonstrating the step of subtracting 3 from both sides to solve for the variable x, with the subtracted '3's highlighted in red.], alt: "The equation x + 3 - 3 = 8 - 3, demonstrating the step of subtracting 3 from both sides to solve for the variable x, with the subtracted '3's highlighted in red.", caption: none)],
  [Then we were left with five.], [#figure(figph[The image displays the mathematical equation 'x = 5' in black text on a plain white background.], alt: "The image displays the mathematical equation 'x = 5' in black text on a plain white background.", caption: none)],
))

Check:

Five in the envelope plus three more does equal eight!

#math.equation(block: true, alt: "5 plus 3 equals 8")[$5 + 3 = 8$]Our model has given us an idea of what we need to do to solve one kind of equation. The goal is to isolate the variable by itself on one side of the equation. To solve equations such as these mathematically, we use the #strong[Subtraction Property of Equality].

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Subtraction Property of Equality]

For any numbers #emph[a], #emph[b], and #emph[c],

#math.equation(block: true, alt: "If, a, equals, b ,; then, a minus c, equals, b minus c")[$"If" &  & a & = & b , \ "then" &  & a − c & = & b − c$]When you subtract the same quantity from both sides of an equation, you still have equality.
]

Let’s see how to use this property to solve an equation. Remember, the goal is to isolate the variable on one side of the equation. And we check our solutions by substituting the value into the equation to make sure we have a true statement.

#examplebox("Example 2")[][
Solve: #math.equation(block: false, alt: "y plus 37 equals −13 .")[$y + 37 = −13 .$]

#solutionbox[
To get #emph[y] by itself, we will undo the addition of 37 by using the Subtraction Property of Equality.

#figure(table(
  columns: 3, align: left, inset: 6pt,
  [], [#figure(figph[A mathematical equation is displayed: y + 37 = -13, which is an algebraic expression involving a variable, addition, and negative numbers.], alt: "A mathematical equation is displayed: y + 37 = -13, which is an algebraic expression involving a variable, addition, and negative numbers.", caption: none)], [],
  [Subtract 37 from each side to ‘undo’ the addition.], [#figure(figph[An equation showing 37 being subtracted from both sides to solve for y: y + 37 - 37 = -13 - 37. The subtractions are highlighted in red.], alt: "An equation showing 37 being subtracted from both sides to solve for y: y + 37 - 37 = -13 - 37. The subtractions are highlighted in red.", caption: none)], [],
  [Simplify.], [#figure(figph[The image displays a mathematical equation: y = -50. The equation is presented in a simple, clear font against a white background.], alt: "The image displays a mathematical equation: y = -50. The equation is presented in a simple, clear font against a white background.", caption: none)], [],
  [Check:], [#figure(figph[A mathematical equation is displayed on a white background: y + 37 = -13. The equation involves a variable 'y', addition, a positive integer, an equality sign, and a negative integer.], alt: "A mathematical equation is displayed on a white background: y + 37 = -13. The equation involves a variable 'y', addition, a positive integer, an equality sign, and a negative integer.", caption: none)], [],
  [Substitute #math.equation(block: false, alt: "y equals −50")[$y = −50$]], [#figure(figph[A mathematical equation displays -50 + 37 = -13, with the -50 term highlighted in a red font, making it stand out from the other numbers and symbols which are in a standard gray.], alt: "A mathematical equation displays -50 + 37 = -13, with the -50 term highlighted in a red font, making it stand out from the other numbers and symbols which are in a standard gray.", caption: none)], [],
  [], [#figure(figph[A mathematical equation shows -13 compared to -13 with a question mark over the equals sign, followed by a checkmark indicating they are indeed equal.], alt: "A mathematical equation shows -13 compared to -13 with a question mark over the equals sign, followed by a checkmark indicating they are indeed equal.", caption: none)], [],
))

Since #math.equation(block: false, alt: "y equals −50")[$y = −50$] makes #math.equation(block: false, alt: "y plus 37 equals −13")[$y + 37 = −13$] a true statement, we have the solution to this equation.
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Solve: #math.equation(block: false, alt: "x plus 19 equals −27")[$x + 19 = −27$].

#solutionbox[
#math.equation(block: true, alt: "x equals −46")[$x = −46$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Solve: #math.equation(block: false, alt: "x plus 16 equals −34")[$x + 16 = −34$].

#solutionbox[
#math.equation(block: true, alt: "x equals −50")[$x = −50$]
]
]

What happens when an equation has a number subtracted from the variable, as in the equation #math.equation(block: false, alt: "x minus 5 equals 8")[$x − 5 = 8$]? We use another property of equations to solve equations where a number is subtracted from the variable. We want to isolate the variable, so to ‘undo’ the subtraction we will add the number to both sides. We use the #strong[Addition Property of Equality].

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Addition Property of Equality]

For any numbers #emph[a], #emph[b], and #emph[c],

#math.equation(block: true, alt: "If, a, equals, b ,; then, a plus c, equals, b plus c")[$"If" &  & a & = & b , \ "then" &  & a + c & = & b + c$]When you add the same quantity to both sides of an equation, you still have equality.
]

In , 37 was added to the #emph[y] and so we subtracted 37 to ‘undo’ the addition. In , we will need to ‘undo’ subtraction by using the #strong[Addition Property of Equality].

#examplebox("Example 3")[][
Solve: #math.equation(block: false, alt: "a minus 28 equals −37 .")[$a − 28 = −37 .$]

#solutionbox[
#figure(table(
  columns: 3, align: left, inset: 6pt,
  [], [#figure(figph[A mathematical equation is displayed with a variable 'a' being subtracted by 28, which equals -37. The equation reads as 'a - 28 = -37'.], alt: "A mathematical equation is displayed with a variable 'a' being subtracted by 28, which equals -37. The equation reads as 'a - 28 = -37'.", caption: none)], [],
  [Add 28 to each side to ‘undo’ the subtraction.], [#figure(figph[Mathematical equation: 'a - 28 + 28 = -37 + 28', showing 28 added to both sides to solve for 'a', with the added numbers in red.], alt: "Mathematical equation: 'a - 28 + 28 = -37 + 28', showing 28 added to both sides to solve for 'a', with the added numbers in red.", caption: none)], [],
  [Simplify.], [#figure(figph[The image displays the mathematical expression 'a = -9' in black text against a white background.], alt: "The image displays the mathematical expression 'a = -9' in black text against a white background.", caption: none)], [],
  [Check:], [#figure(figph[A mathematical equation is displayed, showing 'q - 28 = -37' against a plain white background. The variable 'q' is being solved for in this linear equation.], alt: "A mathematical equation is displayed, showing 'q - 28 = -37' against a plain white background. The variable 'q' is being solved for in this linear equation.", caption: none)], [],
  [Substitute #math.equation(block: false, alt: "a equals −9")[$a = −9$]], [#figure(figph[A mathematical equation shows -9 minus 28 equals -37. The -9 is highlighted in red, while the rest of the numbers and symbols are in black, set against a plain white background.], alt: "A mathematical equation shows -9 minus 28 equals -37. The -9 is highlighted in red, while the rest of the numbers and symbols are in black, set against a plain white background.", caption: none)], [],
  [], [#figure(figph[An equation showing '-37 ?= -37' is resolved with a checkmark, confirming equality.], alt: "An equation showing '-37 ?= -37' is resolved with a checkmark, confirming equality.", caption: none)], [],
  [], [The solution to #math.equation(block: false, alt: "a minus 28 equals −37")[$a − 28 = −37$] is #math.equation(block: false, alt: "a equals −9 .")[$a = −9 .$]], [],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Solve: #math.equation(block: false, alt: "n minus 61 equals −75 .")[$n − 61 = −75 .$]

#solutionbox[
#math.equation(block: true, alt: "n equals −14")[$n = −14$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Solve: #math.equation(block: false, alt: "p minus 41 equals −73 .")[$p − 41 = −73 .$]

#solutionbox[
#math.equation(block: true, alt: "p equals −32")[$p = −32$]
]
]

#examplebox("Example 4")[][
Solve: #math.equation(block: false, alt: "x minus the fraction 5 over 8 equals the fraction 3 over 4 .")[$x − frac(5, 8) = frac(3, 4) .$]

#solutionbox[
#figure(table(
  columns: 3, align: left, inset: 6pt,
  [], [#figure(figph[A mathematical equation shows 'x - 5/8 = 3/4' on a white background.], alt: "A mathematical equation shows 'x - 5/8 = 3/4' on a white background.", caption: none)], [],
  [Use the Addition Property of Equality.], [#figure(figph[An algebraic equation is shown: x - 5/8 + 5/8 = 3/4 + 5/8. The fractions 5/8 appear in red text.], alt: "An algebraic equation is shown: x - 5/8 + 5/8 = 3/4 + 5/8. The fractions 5/8 appear in red text.", caption: none)], [],
  [Find the LCD to add the fractions on the right.], [#figure(figph[A mathematical equation showing x minus 5/8 plus 5/8 equals 6/8 plus 5/8. This algebraic expression requires solving for the variable x, involving basic fraction addition and subtraction.], alt: "A mathematical equation showing x minus 5/8 plus 5/8 equals 6/8 plus 5/8. This algebraic expression requires solving for the variable x, involving basic fraction addition and subtraction.", caption: none)], [],
  [Simplify.], [#figure(figph[The image shows the mathematical equation x equals 11 over 8.], alt: "The image shows the mathematical equation x equals 11 over 8.", caption: none)], [],
  [Check:], [#figure(figph[A mathematical equation is displayed, showing 'x minus five-eighths equals three-fourths' (x - 5/8 = 3/4).], alt: "A mathematical equation is displayed, showing 'x minus five-eighths equals three-fourths' (x - 5/8 = 3/4).", caption: none)], [],
  [Substitute #math.equation(block: false, alt: "x equals the fraction 11 over 8 .")[$x = frac(11, 8) .$]], [#figure(figph[A mathematical equation displaying 11/8 - 5/8 = ? 3/4, where the question mark needs to be replaced by the correct relational operator.], alt: "A mathematical equation displaying 11/8 - 5/8 = ? 3/4, where the question mark needs to be replaced by the correct relational operator.", caption: none)], [],
  [Subtract.], [#figure(figph[A mathematical expression displaying the fraction 6/8 and 3/4, with a question mark over an equals sign, asking if the two fractions are equivalent. Both fractions are indeed equal.], alt: "A mathematical expression displaying the fraction 6/8 and 3/4, with a question mark over an equals sign, asking if the two fractions are equivalent. Both fractions are indeed equal.", caption: none)], [],
  [Simplify.], [#figure(figph[A mathematical expression shows the fraction 3/4 equals 3/4, followed by a checkmark, indicating correctness or approval.], alt: "A mathematical expression shows the fraction 3/4 equals 3/4, followed by a checkmark, indicating correctness or approval.", caption: none)], [],
  [], [The solution to #math.equation(block: false, alt: "x minus the fraction 5 over 8 equals the fraction 3 over 4")[$x − frac(5, 8) = frac(3, 4)$] is #math.equation(block: false, alt: "x equals the fraction 11 over 8 .")[$x = frac(11, 8) .$]], [],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Solve: #math.equation(block: false, alt: "p minus the fraction 2 over 3 equals the fraction 5 over 6 .")[$p − frac(2, 3) = frac(5, 6) .$]

#solutionbox[
#math.equation(block: true, alt: "p equals the fraction 9 over 6 equals the fraction 3 over 2")[$p = frac(9, 6) = frac(3, 2)$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Solve: #math.equation(block: false, alt: "q minus the fraction 1 over 2 equals the fraction 5 over 6 .")[$q − frac(1, 2) = frac(5, 6) .$]

#solutionbox[
#math.equation(block: true, alt: "q equals the fraction 4 over 3")[$q = frac(4, 3)$]
]
]

The next example will be an equation with decimals.

#examplebox("Example 5")[][
Solve: #math.equation(block: false, alt: "n minus 0.63 equals −4.2 .")[$n − 0.63 = −4.2 .$]

#solutionbox[
#figure(table(
  columns: 3, align: left, inset: 6pt,
  [], [#figure(figph[A mathematical equation is displayed with the variable 'n' being subtracted by 0.63, equaling -4.2. The equation reads as 'n - 0.63 = -4.2'.], alt: "A mathematical equation is displayed with the variable 'n' being subtracted by 0.63, equaling -4.2. The equation reads as 'n - 0.63 = -4.2'.", caption: none)], [],
  [Use the Addition Property of Equality.], [#figure(figph[A mathematical equation is displayed, showing 'n - 0.63 + 0.63 = -4.2 + 0.63', with the addition of 0.63 on both sides highlighted in red.], alt: "A mathematical equation is displayed, showing 'n - 0.63 + 0.63 = -4.2 + 0.63', with the addition of 0.63 on both sides highlighted in red.", caption: none)], [],
  [Add.], [#figure(figph[A white background with the text 'n = -3.57' displayed in black font.], alt: "A white background with the text 'n = -3.57' displayed in black font.", caption: none)], [],
  [Check:], [#figure(figph[The image displays the mathematical expression 'n = -3.57' in black text against a plain white background, indicating a numerical value for the variable 'n'.], alt: "The image displays the mathematical expression 'n = -3.57' in black text against a plain white background, indicating a numerical value for the variable 'n'.", caption: none)], [],
  [Let #math.equation(block: false, alt: "n equals −3.57")[$n = −3.57$].], [#figure(figph[A mathematical equation displaying '-3.57 - 0.63 = -4.2' with a question mark above the equal sign, verifying the statement.], alt: "A mathematical equation displaying '-3.57 - 0.63 = -4.2' with a question mark above the equal sign, verifying the statement.", caption: none)], [],
  [], [#figure(figph[The mathematical equation -4.2 = -4.2 is displayed, confirmed as correct with a checkmark.], alt: "The mathematical equation -4.2 = -4.2 is displayed, confirmed as correct with a checkmark.", caption: none)], [],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Solve: #math.equation(block: false, alt: "b minus 0.47 equals −2.1 .")[$b − 0.47 = −2.1 .$]

#solutionbox[
#math.equation(block: true, alt: "b equals −1.63")[$b = −1.63$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Solve: #math.equation(block: false, alt: "c minus 0.93 equals −4.6 .")[$c − 0.93 = −4.6 .$]

#solutionbox[
#math.equation(block: true, alt: "c equals −3.67")[$c = −3.67$]
]
]


=== Solve Equations That Require Simplification

In the previous examples, we were able to isolate the variable with just one operation. Most of the equations we encounter in algebra will take more steps to solve. Usually, we will need to simplify one or both sides of an equation before using the Subtraction or Addition Properties of Equality.

You should always simplify as much as possible before you try to isolate the variable. Remember that to simplify an expression means to do all the operations in the expression. Simplify one side of the equation at a time. Note that #strong[simplification] is different from the process used to solve an equation in which we apply an operation to both sides.

#examplebox("Example 6")[How to Solve Equations That Require Simplification][
Solve: #math.equation(block: false, alt: "9 x minus 5 minus 8 x minus 6 equals 7 .")[$9 x − 5 − 8 x − 6 = 7 .$]

#solutionbox[
#figure(figph[This figure is a table that has three columns and four rows. The first column is a header column, and it contains the names and numbers of each step. The second column contains further written instructions. The third column contains math. On the top row of the table, the first cell on the left reads: “Step 1. Simplify the expressions on each side as much as possible.” The text in the second cell reads: “Rearrange the terms, using the Commutative Property of Addition. Combine like terms. Notice that each side is now simplified as much as possible.” The third cell contains the equation 9 x minus 5 minus 8 x minus 6 equals 7. Below this is the same equation, with the terms rearranged: 9 x minus 8 x minus 5 minus 6 equals 7. Below this is the equation with like terms combined: x minus 11 equals 7.], alt: "This figure is a table that has three columns and four rows. The first column is a header column, and it contains the names and numbers of each step. The second column contains further written instructions. The third column contains math. On the top row of the table, the first cell on the left reads: “Step 1. Simplify the expressions on each side as much as possible.” The text in the second cell reads: “Rearrange the terms, using the Commutative Property of Addition. Combine like terms. Notice that each side is now simplified as much as possible.” The third cell contains the equation 9 x minus 5 minus 8 x minus 6 equals 7. Below this is the same equation, with the terms rearranged: 9 x minus 8 x minus 5 minus 6 equals 7. Below this is the equation with like terms combined: x minus 11 equals 7.", caption: none)

#figure(figph[In the second row of the table, the first cell says “Step 2. Isolate the variable.” In the second cell, the instructions say “Now isolate x. Undo subtraction by adding 11 to both sides.” The third cell contains the equation x minus 11 plus 11 equals 7 plus 11, with “plus 11” written in red on both sides.], alt: "In the second row of the table, the first cell says “Step 2. Isolate the variable.” In the second cell, the instructions say “Now isolate x. Undo subtraction by adding 11 to both sides.” The third cell contains the equation x minus 11 plus 11 equals 7 plus 11, with “plus 11” written in red on both sides.", caption: none)

#figure(figph[In the third row of the table, the first cell says: “Step 3. Simplify the equation on both sides of the equation.” The second cell is left blank. The third cell contains x equals 18.], alt: "In the third row of the table, the first cell says: “Step 3. Simplify the equation on both sides of the equation.” The second cell is left blank. The third cell contains x equals 18.", caption: none)

#figure(figph[In the fourth and bottom row of the table, the first cell says: “Step 4. Check the solution.” The second cell is blank. In the third cell is the text “Check: Substitute x equals 18.” Below this is the equation 9 x minus 5 minus 8 x minus 6 equals 7. Underneath is the same equation, with 18 written in red in parentheses replacing each x: 9 times 18 (in parentheses) minus 5 minus 8 times 18 (in parentheses) minus 6 might equal 7. Below is the equation 162 minus 5 minus 144 minus 6 might equal 7. Below this is the equation 157 minus 144 minus 6 might equal 7. Below this is 13 minus 6 might equal 7. On the last line is the equation 7 equals 7, with a check mark next to it.], alt: "In the fourth and bottom row of the table, the first cell says: “Step 4. Check the solution.” The second cell is blank. In the third cell is the text “Check: Substitute x equals 18.” Below this is the equation 9 x minus 5 minus 8 x minus 6 equals 7. Underneath is the same equation, with 18 written in red in parentheses replacing each x: 9 times 18 (in parentheses) minus 5 minus 8 times 18 (in parentheses) minus 6 might equal 7. Below is the equation 162 minus 5 minus 144 minus 6 might equal 7. Below this is the equation 157 minus 144 minus 6 might equal 7. Below this is 13 minus 6 might equal 7. On the last line is the equation 7 equals 7, with a check mark next to it.", caption: none)
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Solve: #math.equation(block: false, alt: "8 y minus 4 minus 7 y minus 7 equals 4 .")[$8 y − 4 − 7 y − 7 = 4 .$]

#solutionbox[
#math.equation(block: true, alt: "y equals 15")[$y = 15$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Solve: #math.equation(block: false, alt: "6 z plus 5 minus 5 z minus 4 equals 3 .")[$6 z + 5 − 5 z − 4 = 3 .$]

#solutionbox[
#math.equation(block: true, alt: "z equals 2")[$z = 2$]
]
]

#examplebox("Example 7")[][
Solve: #math.equation(block: false, alt: "5 open parenthesis n minus 4 close parenthesis minus 4 n equals −8 .")[$5 ( n − 4 ) − 4 n = −8 .$]

#solutionbox[
We simplify both sides of the equation as much as possible before we try to isolate the variable.

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#figure(figph[A mathematical equation is displayed, showing 5 multiplied by the quantity (n - 4), minus 4n, which equals -8. The equation is 5(n - 4) - 4n = -8.], alt: "A mathematical equation is displayed, showing 5 multiplied by the quantity (n - 4), minus 4n, which equals -8. The equation is 5(n - 4) - 4n = -8.", caption: none)]),
  [Distribute on the left.], [#figure(figph[A mathematical equation on a white background reads: 5n - 20 - 4n = -8.], alt: "A mathematical equation on a white background reads: 5n - 20 - 4n = -8.", caption: none)],
  [Use the Commutative Property to rearrange terms.], [#figure(figph[The equation 5n - 4n - 20 = -8 is shown.], alt: "The equation 5n - 4n - 20 = -8 is shown.", caption: none)],
  [Combine like terms.], [#figure(figph[The mathematical equation n - 20 = -8 is displayed in black text on a plain white background.], alt: "The mathematical equation n - 20 = -8 is displayed in black text on a plain white background.", caption: none)],
  [Each side is as simplified as possible. Next, isolate #math.equation(block: false, alt: "n")[$n$].], [],
  [Undo subtraction by using the Addition Property of Equality.], [#figure(figph[A mathematical equation shows 'n - 20 + 20 = -8 + 20'. The numbers '+20' are highlighted in red on both sides of the equation, indicating that 20 is being added to both sides to solve for 'n'.], alt: "A mathematical equation shows 'n - 20 + 20 = -8 + 20'. The numbers '+20' are highlighted in red on both sides of the equation, indicating that 20 is being added to both sides to solve for 'n'.", caption: none)],
  [Add.], [#figure(figph[The image displays the text 'n = 12' in black, centrally placed against a plain white background.], alt: "The image displays the text 'n = 12' in black, centrally placed against a plain white background.", caption: none)],
  [Check. Substitute #math.equation(block: false, alt: "n equals 12")[$n = 12$]. #linebreak() #figure(figph[Verification of the algebraic equation 5(n-4)-4n=-8. Substituting n=12 into the equation simplifies to -8=-8, confirming that n=12 is the correct solution.], alt: "Verification of the algebraic equation 5(n-4)-4n=-8. Substituting n=12 into the equation simplifies to -8=-8, confirming that n=12 is the correct solution.", caption: none)], [],
  [], [The solution to #math.equation(block: false, alt: "5 open parenthesis n minus 4 close parenthesis minus 4 n equals −8")[$5 ( n − 4 ) − 4 n = −8$] is #math.equation(block: false, alt: "n equals 12 .")[$n = 12 .$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Solve: #math.equation(block: false, alt: "5 open parenthesis p minus 3 close parenthesis minus 4 p equals −10 .")[$5 ( p − 3 ) − 4 p = −10 .$]

#solutionbox[
#math.equation(block: true, alt: "p equals 5")[$p = 5$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Solve: #math.equation(block: false, alt: "4 open parenthesis q plus 2 close parenthesis minus 3 q equals −8 .")[$4 ( q + 2 ) − 3 q = −8 .$]

#solutionbox[
#math.equation(block: true, alt: "q equals −16")[$q = −16$]
]
]

#examplebox("Example 8")[][
Solve: #math.equation(block: false, alt: "3 open parenthesis 2 y minus 1 close parenthesis minus 5 y equals 2 open parenthesis y plus 1 close parenthesis minus 2 open parenthesis y plus 3 close parenthesis .")[$3 ( 2 y − 1 ) − 5 y = 2 ( y + 1 ) − 2 ( y + 3 ) .$]

#solutionbox[
We simplify both sides of the equation before we isolate the variable.

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#figure(figph[An algebraic equation is shown, featuring the variable 'y' on both sides: 3(2y - 1) - 5y = 2(y + 1) - 2(y + 3).], alt: "An algebraic equation is shown, featuring the variable 'y' on both sides: 3(2y - 1) - 5y = 2(y + 1) - 2(y + 3).", caption: none)]),
  [Distribute on both sides.], [#figure(figph[A mathematical equation is displayed on a white background. The equation reads: 6y - 3 - 5y = 2y + 2 - 2y - 6.], alt: "A mathematical equation is displayed on a white background. The equation reads: 6y - 3 - 5y = 2y + 2 - 2y - 6.", caption: none)],
  [Use the Commutative Property of Addition.], [#figure(figph[A mathematical equation is displayed: 6y - 5y - 3 = 2y - 2y + 2 - 6, showing terms with the variable 'y' and constant numbers on both sides of the equality.], alt: "A mathematical equation is displayed: 6y - 5y - 3 = 2y - 2y + 2 - 6, showing terms with the variable 'y' and constant numbers on both sides of the equality.", caption: none)],
  [Combine like terms.], [#figure(figph[A mathematical equation displayed on a white background, showing 'y - 3 = -4' in a clear, dark gray font.], alt: "A mathematical equation displayed on a white background, showing 'y - 3 = -4' in a clear, dark gray font.", caption: none)],
  [Each side is as simplified as possible. Next, isolate #math.equation(block: false, alt: "y")[$y$].], [],
  [Undo subtraction by using the Addition Property of Equality.], [#figure(figph[An algebraic equation: y - 3 + 3 = -4 + 3. The '+3' terms on both sides are highlighted in red, demonstrating the addition property of equality to isolate the variable y.], alt: "An algebraic equation: y - 3 + 3 = -4 + 3. The '+3' terms on both sides are highlighted in red, demonstrating the addition property of equality to isolate the variable y.", caption: none)],
  [Add.], [#figure(figph[The mathematical equation y = -1 is displayed in black text against a plain white background.], alt: "The mathematical equation y = -1 is displayed in black text against a plain white background.", caption: none)],
  [Check. Let #math.equation(block: false, alt: "y equals −1")[$y = −1$]. #linebreak() #figure(figph[Verifying an algebraic solution. This image shows the step-by-step process of substituting y = -1 into the equation 3(2y-1)-5y = 2(y+1)-2(y+3), confirming that -1 is the correct value as both sides equal -4.], alt: "Verifying an algebraic solution. This image shows the step-by-step process of substituting y = -1 into the equation 3(2y-1)-5y = 2(y+1)-2(y+3), confirming that -1 is the correct value as both sides equal -4.", caption: none)], [],
  [], [The solution to #math.equation(block: false, alt: "3 open parenthesis 2 y minus 1 close parenthesis minus 5 y equals 2 open parenthesis y plus 1 close parenthesis minus 2 open parenthesis y plus 3 close parenthesis")[$3 ( 2 y − 1 ) − 5 y = 2 ( y + 1 ) − 2 ( y + 3 )$] is #math.equation(block: false, alt: "y equals −1 .")[$y = −1 .$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Solve: #math.equation(block: false, alt: "4 open parenthesis 2 h minus 3 close parenthesis minus 7 h equals 6 open parenthesis h minus 2 close parenthesis minus 6 open parenthesis h minus 1 close parenthesis .")[$4 ( 2 h − 3 ) − 7 h = 6 ( h − 2 ) − 6 ( h − 1 ) .$]

#solutionbox[
#math.equation(block: true, alt: "h equals 6")[$h = 6$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Solve: #math.equation(block: false, alt: "2 open parenthesis 5 x plus 2 close parenthesis minus 9 x equals 3 open parenthesis x minus 2 close parenthesis minus 3 open parenthesis x minus 4 close parenthesis .")[$2 ( 5 x + 2 ) − 9 x = 3 ( x − 2 ) − 3 ( x − 4 ) .$]

#solutionbox[
#math.equation(block: true, alt: "x equals 2")[$x = 2$]
]
]


=== Translate to an Equation and Solve

To solve applications algebraically, we will begin by translating from English sentences into equations. Our first step is to look for the word (or words) that would translate to the #strong[equals sign].  shows us some of the words that are commonly used.

#figure(table(
  columns: 1, align: left, inset: 6pt,
  [#strong[Equals =]],
  [is #linebreak() is equal to #linebreak() is the same as #linebreak() the result is #linebreak() gives #linebreak() was #linebreak() will be],
))

The steps we use to translate a sentence into an equation are listed below.

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Translate an English sentence to an algebraic equation.]

+ Locate the “equals” word(s). Translate to an equals sign (=).
+ Translate the words to the left of the “equals” word(s) into an algebraic expression.
+ Translate the words to the right of the “equals” word(s) into an algebraic expression.
]

#examplebox("Example 9")[][
Translate and solve: Eleven more than #emph[x] is equal to 54.

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([Translate.], [#figure(figph[An image illustrating the translation of the phrase 'Eleven more than x is equal to 54' into the algebraic equation 'x + 11 = 54', with corresponding parts underlined.], alt: "An image illustrating the translation of the phrase 'Eleven more than x is equal to 54' into the algebraic equation 'x + 11 = 54', with corresponding parts underlined.", caption: none)]),
  [Subtract 11 from both sides.], [#figure(figph[The equation 'x + 11 - 11 = 54 - 11' demonstrates a step in solving for 'x' by applying the subtraction property of equality to both sides.], alt: "The equation 'x + 11 - 11 = 54 - 11' demonstrates a step in solving for 'x' by applying the subtraction property of equality to both sides.", caption: none)],
  [Simplify.], [#figure(figph[A close-up shot of a white surface with the mathematical equation 'X = 43' written in black characters, likely indicating a simple algebraic problem or a labeled value.], alt: "A close-up shot of a white surface with the mathematical equation 'X = 43' written in black characters, likely indicating a simple algebraic problem or a labeled value.", caption: none)],
  [Check: Is 54 eleven more than 43? #linebreak() 
    #math.equation(block: false, alt: "43 plus 11, equals ?, 54; 54, equals, 54 ✓")[$43 + 11 & limits(=)^(?) & 54 \ 54 & = & 54 ✓$]], [],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Translate and solve: Ten more than #emph[x] is equal to 41.

#solutionbox[
#math.equation(block: true, alt: "x plus 10 equals 41 ; x equals 31")[$x + 10 = 41 ; x = 31$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Translate and solve: Twelve less than #emph[x] is equal to 51.

#solutionbox[
#math.equation(block: true, alt: "x minus 12 equals 51 ; x equals 63")[$x − 12 = 51 ; x = 63$]
]
]

#examplebox("Example 10")[][
Translate and solve: The difference of #math.equation(block: false, alt: "12 t")[$12 t$] and #math.equation(block: false, alt: "11 t")[$11 t$] is #math.equation(block: false, alt: "−14")[$−14$].

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([Translate.], [#figure(figph[This image illustrates how to translate a word problem into an algebraic equation. 'The difference of 12t and 11t is -14' translates to 12t - 11t = -14.], alt: "This image illustrates how to translate a word problem into an algebraic equation. 'The difference of 12t and 11t is -14' translates to 12t - 11t = -14.", caption: none)]),
  [Simplify.], [#figure(figph[The text 't = -14' is displayed in the top right corner against a plain white background.], alt: "The text 't = -14' is displayed in the top right corner against a plain white background.", caption: none)],
  [Check: #linebreak() 
    #math.equation(block: false, alt: "12 open parenthesis −14 close parenthesis minus 11 open parenthesis −14 close parenthesis, equals ?, −14; −168 plus 154, equals ?, −14; −14, equals, −14 ✓")[$12 ( −14 ) − 11 ( −14 ) & limits(=)^(?) & −14 \ −168 + 154 & limits(=)^(?) & −14 \ −14 & = & −14 ✓$]], [],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Translate and solve: The difference of #math.equation(block: false, alt: "4 x")[$4 x$] and #math.equation(block: false, alt: "3 x")[$3 x$] is 14.

#solutionbox[
#math.equation(block: true, alt: "4 x minus 3 x equals 14 ; x equals 14")[$4 x − 3 x = 14 ; x = 14$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Translate and solve: The difference of #math.equation(block: false, alt: "7 a")[$7 a$] and #math.equation(block: false, alt: "6 a")[$6 a$] is #math.equation(block: false, alt: "−8")[$−8$].

#solutionbox[
#math.equation(block: true, alt: "7 a minus 6 a equals −8 ; a equals −8")[$7 a − 6 a = −8 ; a = −8$]
]
]


=== Translate and Solve Applications

Most of the time a question that requires an algebraic solution comes out of a real life question. To begin with that question is asked in English (or the language of the person asking) and not in math symbols. Because of this, it is an important skill to be able to translate an everyday situation into algebraic language.

We will start by restating the problem in just one sentence, assign a variable, and then translate the sentence into an equation to solve. When assigning a variable, choose a letter that reminds you of what you are looking for. For example, you might use #emph[q] for the number of quarters if you were solving a problem about coins.

#examplebox("Example 11")[How to Solve Translate and Solve Applications][
The MacIntyre family recycled newspapers for two months. The two months of newspapers weighed a total of 57 pounds. The second month, the newspapers weighed 28 pounds. How much did the newspapers weigh the first month?

#solutionbox[
#figure(figph[This figure is a table that has three columns and four rows. The first column is a header column, and it contains the names and numbers of each step. The second column contains further written instructions. The third column contains text and algebra. In the top row, the first cell says “Step 1. Read the problem. Make sure all the words and ideas are understood.” The text in the second cell says “The problem is about the weight of newspapers.” The third cell is blank.], alt: "This figure is a table that has three columns and four rows. The first column is a header column, and it contains the names and numbers of each step. The second column contains further written instructions. The third column contains text and algebra. In the top row, the first cell says “Step 1. Read the problem. Make sure all the words and ideas are understood.” The text in the second cell says “The problem is about the weight of newspapers.” The third cell is blank.", caption: none)

#figure(figph[In the second row, the first cell says “Step 2. Identify what we are asked to find.” The second cell says “What are we asked to find?” The third cell says: “How much did the newspapers weigh the 2nd month?”], alt: "In the second row, the first cell says “Step 2. Identify what we are asked to find.” The second cell says “What are we asked to find?” The third cell says: “How much did the newspapers weigh the 2nd month?”", caption: none)

#figure(figph[In the third row, the first cell says “Step 3. Name what we are looking for. Choose a variable to represent that quantity.” The second cell says “Choose a variable.” The third cell says “Let w equal weight of the newspapers the 1st month.”], alt: "In the third row, the first cell says “Step 3. Name what we are looking for. Choose a variable to represent that quantity.” The second cell says “Choose a variable.” The third cell says “Let w equal weight of the newspapers the 1st month.”", caption: none)

#figure(figph[In the fourth row, the first cell says “Step 4. Translate into an equation. It may be helpful to restate the problem in one sentence with the important information.” The second cell says “Restate the problem. We know that the weight of the newspapers the second month is 28 pounds.” The third cell says “Weight of newspapers the 1st month plus the weight of the newspapers the 2nd month equals 57 pounds. Weight from 1st month plus 28 equals 57.” One line down, the second cell says “Translate into an equation using the variable w.” The third cell contains the equation w plus 28 equals 57.], alt: "In the fourth row, the first cell says “Step 4. Translate into an equation. It may be helpful to restate the problem in one sentence with the important information.” The second cell says “Restate the problem. We know that the weight of the newspapers the second month is 28 pounds.” The third cell says “Weight of newspapers the 1st month plus the weight of the newspapers the 2nd month equals 57 pounds. Weight from 1st month plus 28 equals 57.” One line down, the second cell says “Translate into an equation using the variable w.” The third cell contains the equation w plus 28 equals 57.", caption: none)

#figure(figph[In the fifth row, the first cell says “Step 5. Solve the equation using good algebra techniques.” The second cell says “Solve.” The third cell contains the equation with 28 being subtracted from both sides: w plus 28 minus 28 equals 57 minus 28, with minus 28 written in red. Below this is w equals 29.], alt: "In the fifth row, the first cell says “Step 5. Solve the equation using good algebra techniques.” The second cell says “Solve.” The third cell contains the equation with 28 being subtracted from both sides: w plus 28 minus 28 equals 57 minus 28, with minus 28 written in red. Below this is w equals 29.", caption: none)

#figure(figph[In the sixth row, the first cell says “Step 6. Check the answer and make sure it makes sense.” The second cell says “Does 1st month’s weight plus 2nd month’s weight equal 57 pounds?” The third cell contains the equation 29 plus 28 might equal 57. Below this is 57 equals 57 with a check mark next to it.], alt: "In the sixth row, the first cell says “Step 6. Check the answer and make sure it makes sense.” The second cell says “Does 1st month’s weight plus 2nd month’s weight equal 57 pounds?” The third cell contains the equation 29 plus 28 might equal 57. Below this is 57 equals 57 with a check mark next to it.", caption: none)

#figure(figph[In the seventh and final row, the first cell says ‘Step 7. Answer the question with a complete sentence.” The second cell says “Write a sentence to answer ‘How much did the newspapers weigh the 2nd month?’” The third cell contains the sentence “The 2nd month the newspapers weighed 29 pounds.”], alt: "In the seventh and final row, the first cell says ‘Step 7. Answer the question with a complete sentence.” The second cell says “Write a sentence to answer ‘How much did the newspapers weigh the 2nd month?’” The third cell contains the sentence “The 2nd month the newspapers weighed 29 pounds.”", caption: none)
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Translate into an algebraic equation and solve:

The Pappas family has two cats, Zeus and Athena. Together, they weigh 23 pounds. Zeus weighs 16 pounds. How much does Athena weigh?

#solutionbox[
7 pounds
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Translate into an algebraic equation and solve:

Sam and Henry are roommates. Together, they have 68 books. Sam has 26 books. How many books does Henry have?

#solutionbox[
42 books
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Solve an application.]

+ #strong[Read] the problem. Make sure all the words and ideas are understood.
+ #strong[Identify] what we are looking for.
+ #strong[Name] what we are looking for. Choose a variable to represent that quantity.
+ #strong[Translate] into an equation. It may be helpful to restate the problem in one sentence with the important information.
+ #strong[Solve] the equation using good algebra techniques.
+ #strong[Check] the answer in the problem and make sure it makes sense.
+ #strong[Answer] the question with a complete sentence.
]

#examplebox("Example 12")[][
Randell paid \$28,675 for his new car. This was \$875 less than the sticker price. What was the sticker price of the car?

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([#emph[Step 1. Read] the problem.], []),
  [#emph[Step 2. Identify] what we are looking for.], ["What was the sticker price of the car?"],
  [#emph[Step 3. Name] what we are looking for.
     #linebreak() Choose a variable to represent that quantity.], [Let #math.equation(block: false, alt: "s equals")[$s =$] the sticker price of the car.],
  [#emph[Step 4. Translate] into an equation. Restate the problem in one sentence.], [\$28,675 is \$875 less than the sticker price],
  [#emph[Step 5. Solve] the equation.], [#math.equation(block: false, alt: "$28,675 is $875 less than s")[$"$28,675 is $875 less than" #h(0.2em) s$]
    #math.equation(block: false, alt: "28,675, equals, s minus 875; 28,675 plus 875, equals, s minus 875 plus 875; 29,550, equals, s")[$28,675 & = & s − 875 \ 28,675 + 875 & = & s − 875 + 875 \ 29,550 & = & s$]],
  [#emph[Step 6. Check] the answer.
	 #linebreak() Is \$875 less than \$29,550 equal to \$28,675?
	 #linebreak() #math.equation(block: false, alt: "29,550 minus 875, equals ?, 28,675; 28,675, equals, 28,675 ✓")[$29,550 − 875 & limits(=)^(?) & 28,675 \ 28,675 & = & 28,675 ✓$]], [],
  [#emph[Step 7. Answer] the question with a complete sentence.], [The sticker price of the car was \$29,550.],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Translate into an algebraic equation and solve:

Eddie paid \$19,875 for his new car. This was \$1,025 less than the sticker price. What was the sticker price of the car?

#solutionbox[
\$20,900
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Translate into an algebraic equation and solve:

The admission price for the movies during the day is \$7.75. This is \$3.25 less the price at night. How much does the movie cost at night?

#solutionbox[
\$11.00
]
]


=== Key Concepts

- #strong[To Determine Whether a Number is a Solution to an Equation]
+ #strong[Substitute the number in for the variable in the equation.]
+ #strong[Simplify the expressions on both sides of the equation.]
+ #strong[Determine whether the resulting statement is true.]
- If it is true, the number is a solution.
- If it is not true, the number is not a solution.
- #strong[Addition Property of Equality]
- For any numbers #emph[a], #emph[b], and #emph[c], if #math.equation(block: false, alt: "a equals b")[$a = b$], then #math.equation(block: false, alt: "a plus c equals b plus c")[$a + c = b + c$].
- #strong[Subtraction Property of Equality]
- For any numbers #emph[a], #emph[b], and #emph[c], if #math.equation(block: false, alt: "a equals b")[$a = b$], then #math.equation(block: false, alt: "a minus c equals b minus c")[$a − c = b − c$].
- #strong[To Translate a Sentence to an Equation]
+ Locate the “equals” word(s). Translate to an equal sign (=).
+ Translate the words to the left of the “equals” word(s) into an algebraic expression.
+ Translate the words to the right of the “equals” word(s) into an algebraic expression.
- #strong[To Solve an Application]
+ Read the problem. Make sure all the words and ideas are understood.
+ Identify what we are looking for.
+ Name what we are looking for. Choose a variable to represent that quantity.
+ Translate into an equation. It may be helpful to restate the problem in one sentence with the important information.
+ Solve the equation using good algebra techniques.
+ Check the answer in the problem and make sure it makes sense.
+ Answer the question with a complete sentence.


==== Practice Makes Perfect

#strong[Verify a Solution of an Equation]

In the following exercises, determine whether the given value is a solution to the equation.

Is #math.equation(block: false, alt: "y equals the fraction 5 over 3")[$y = frac(5, 3)$] a solution of #linebreak() #math.equation(block: false, alt: "6 y plus 10 equals 12 y")[$6 y + 10 = 12 y$]?

#solutionbox[
yes
]

Is #math.equation(block: false, alt: "x equals the fraction 9 over 4")[$x = frac(9, 4)$] a solution of #linebreak() #math.equation(block: false, alt: "4 x plus 9 equals 8 x")[$4 x + 9 = 8 x$]?

Is #math.equation(block: false, alt: "u equals minus the fraction 1 over 2")[$u = − frac(1, 2)$] a solution of #linebreak() #math.equation(block: false, alt: "8 u minus 1 equals 6 u")[$8 u − 1 = 6 u$]?

#solutionbox[
no
]

Is #math.equation(block: false, alt: "v equals minus the fraction 1 over 3")[$v = − frac(1, 3)$] a solution of #linebreak() #math.equation(block: false, alt: "9 v minus 2 equals 3 v")[$9 v − 2 = 3 v$]?

#strong[Solve Equations using the Subtraction and Addition Properties of Equality]

In the following exercises, solve each equation using the Subtraction and Addition Properties of Equality.

#math.equation(block: true, alt: "x plus 24 equals 35")[$x + 24 = 35$]

#solutionbox[
#math.equation(block: true, alt: "x equals 11")[$x = 11$]
]

#math.equation(block: true, alt: "x plus 17 equals 22")[$x + 17 = 22$]

#math.equation(block: true, alt: "y plus 45 equals −66")[$y + 45 = −66$]

#solutionbox[
#math.equation(block: true, alt: "y equals −111")[$y = −111$]
]

#math.equation(block: true, alt: "y plus 39 equals −83")[$y + 39 = −83$]

#math.equation(block: true, alt: "b plus the fraction 1 over 4 equals the fraction 3 over 4")[$b + frac(1, 4) = frac(3, 4)$]

#solutionbox[
#math.equation(block: true, alt: "b equals the fraction 1 over 2")[$b = frac(1, 2)$]
]

#math.equation(block: true, alt: "a plus the fraction 2 over 5 equals the fraction 4 over 5")[$a + frac(2, 5) = frac(4, 5)$]

#math.equation(block: true, alt: "p plus 2.4 equals −9.3")[$p + 2.4 = −9.3$]

#solutionbox[
#math.equation(block: true, alt: "p equals −11.7")[$p = −11.7$]
]

#math.equation(block: true, alt: "m plus 7.9 equals 11.6")[$m + 7.9 = 11.6$]

#math.equation(block: true, alt: "a minus 45 equals 76")[$a − 45 = 76$]

#solutionbox[
#math.equation(block: true, alt: "a equals 121")[$a = 121$]
]

#math.equation(block: true, alt: "a minus 30 equals 57")[$a − 30 = 57$]

#math.equation(block: true, alt: "m minus 18 equals −200")[$m − 18 = −200$]

#solutionbox[
#math.equation(block: true, alt: "m equals −182")[$m = −182$]
]

#math.equation(block: true, alt: "m minus 12 equals −12")[$m − 12 = −12$]

#math.equation(block: true, alt: "x minus the fraction 1 over 3 equals 2")[$x − frac(1, 3) = 2$]

#solutionbox[
#math.equation(block: true, alt: "x equals the fraction 7 over 3")[$x = frac(7, 3)$]
]

#math.equation(block: true, alt: "x minus the fraction 1 over 5 equals 4")[$x − frac(1, 5) = 4$]

#math.equation(block: true, alt: "y minus 3.8 equals 10")[$y − 3.8 = 10$]

#solutionbox[
#math.equation(block: true, alt: "y equals 13.8")[$y = 13.8$]
]

#math.equation(block: true, alt: "y minus 7.2 equals 5")[$y − 7.2 = 5$]

#math.equation(block: true, alt: "x minus 165 equals −420")[$x − 165 = −420$]

#solutionbox[
#math.equation(block: true, alt: "x equals −255")[$x = −255$]
]

#math.equation(block: true, alt: "z minus 101 equals −314")[$z − 101 = −314$]

#math.equation(block: true, alt: "z plus 0.52 equals −8.5")[$z + 0.52 = −8.5$]

#solutionbox[
#math.equation(block: true, alt: "z equals −9.02")[$z = −9.02$]
]

#math.equation(block: true, alt: "x plus 0.93 equals −4.1")[$x + 0.93 = −4.1$]

#math.equation(block: true, alt: "q plus the fraction 3 over 4 equals the fraction 1 over 2")[$q + frac(3, 4) = frac(1, 2)$]

#solutionbox[
#math.equation(block: true, alt: "q equals minus the fraction 1 over 4")[$q = − frac(1, 4)$]
]

#math.equation(block: true, alt: "p plus the fraction 1 over 3 equals the fraction 5 over 6")[$p + frac(1, 3) = frac(5, 6)$]

#math.equation(block: true, alt: "p minus the fraction 2 over 5 equals the fraction 2 over 3")[$p − frac(2, 5) = frac(2, 3)$]

#solutionbox[
#math.equation(block: true, alt: "p equals the fraction 16 over 15")[$p = frac(16, 15)$]
]

#math.equation(block: true, alt: "y minus the fraction 3 over 4 equals the fraction 3 over 5")[$y − frac(3, 4) = frac(3, 5)$]

#strong[Solve Equations that Require Simplification]

In the following exercises, solve each equation.

#math.equation(block: true, alt: "c plus 31 minus 10 equals 46")[$c + 31 − 10 = 46$]

#solutionbox[
#math.equation(block: true, alt: "c equals 25")[$c = 25$]
]

#math.equation(block: true, alt: "m plus 16 minus 28 equals 5")[$m + 16 − 28 = 5$]

#math.equation(block: true, alt: "9 x plus 5 minus 8 x plus 14 equals 20")[$9 x + 5 − 8 x + 14 = 20$]

#solutionbox[
#math.equation(block: true, alt: "x equals 1")[$x = 1$]
]

#math.equation(block: true, alt: "6 x plus 8 minus 5 x plus 16 equals 32")[$6 x + 8 − 5 x + 16 = 32$]

#math.equation(block: true, alt: "−6 x minus 11 plus 7 x minus 5 equals −16")[$−6 x − 11 + 7 x − 5 = −16$]

#solutionbox[
#math.equation(block: true, alt: "x equals 0")[$x = 0$]
]

#math.equation(block: true, alt: "−8 n minus 17 plus 9 n minus 4 equals −41")[$−8 n − 17 + 9 n − 4 = −41$]

#math.equation(block: true, alt: "5 open parenthesis y minus 6 close parenthesis minus 4 y equals −6")[$5 ( y − 6 ) − 4 y = −6$]

#solutionbox[
#math.equation(block: true, alt: "y equals 24")[$y = 24$]
]

#math.equation(block: true, alt: "9 open parenthesis y minus 2 close parenthesis minus 8 y equals −16")[$9 ( y − 2 ) − 8 y = −16$]

#math.equation(block: true, alt: "8 open parenthesis u plus 1.5 close parenthesis minus 7 u equals 4.9")[$8 ( u + 1.5 ) − 7 u = 4.9$]

#solutionbox[
#math.equation(block: true, alt: "u equals −7.1")[$u = −7.1$]
]

#math.equation(block: true, alt: "5 open parenthesis w plus 2.2 close parenthesis minus 4 w equals 9.3")[$5 ( w + 2.2 ) − 4 w = 9.3$]

#math.equation(block: true, alt: "6 a minus 5 open parenthesis a minus 2 close parenthesis plus 9 equals −11")[$6 a − 5 ( a − 2 ) + 9 = −11$]

#solutionbox[
#math.equation(block: true, alt: "a equals −30")[$a = −30$]
]

#math.equation(block: true, alt: "8 c minus 7 open parenthesis c minus 3 close parenthesis plus 4 equals −16")[$8 c − 7 ( c − 3 ) + 4 = −16$]

#math.equation(block: true, alt: "6 open parenthesis y minus 2 close parenthesis minus 5 y equals 4 open parenthesis y plus 3 close parenthesis")[$6 ( y − 2 ) − 5 y = 4 ( y + 3 )$] #linebreak() #math.equation(block: true, alt: "minus 4 open parenthesis y minus 1 close parenthesis")[$− 4 ( y − 1 )$]

#solutionbox[
#math.equation(block: true, alt: "y equals 28")[$y = 28$]
]

#math.equation(block: true, alt: "9 open parenthesis x minus 1 close parenthesis minus 8 x equals −3 open parenthesis x plus 5 close parenthesis")[$9 ( x − 1 ) − 8 x = −3 ( x + 5 )$] #linebreak() #math.equation(block: true, alt: "plus 3 open parenthesis x minus 5 close parenthesis")[$+ 3 ( x − 5 )$]

#math.equation(block: true, alt: "3 open parenthesis 5 n minus 1 close parenthesis minus 14 n plus 9")[$3 ( 5 n − 1 ) − 14 n + 9$] #linebreak() #math.equation(block: true, alt: "equals 10 open parenthesis n minus 4 close parenthesis minus 6 n minus 4 open parenthesis n plus 1 close parenthesis")[$= 10 ( n − 4 ) − 6 n − 4 ( n + 1 )$]

#solutionbox[
#math.equation(block: true, alt: "n equals −50")[$n = −50$]
]

#math.equation(block: true, alt: "2 open parenthesis 8 m plus 3 close parenthesis minus 15 m minus 4")[$2 ( 8 m + 3 ) − 15 m − 4$] #linebreak() #math.equation(block: true, alt: "equals 9 open parenthesis m plus 6 close parenthesis minus 2 open parenthesis m minus 1 close parenthesis minus 7 m")[$= 9 ( m + 6 ) − 2 ( m − 1 ) − 7 m$]

#math.equation(block: true, alt: "− open parenthesis j plus 2 close parenthesis plus 2 j minus 1 equals 5")[$"−" ( j + 2 ) + 2 j − 1 = 5$]

#solutionbox[
#math.equation(block: true, alt: "j equals 8")[$j = 8$]
]

#math.equation(block: true, alt: "− open parenthesis k plus 7 close parenthesis plus 2 k plus 8 equals 7")[$"−" ( k + 7 ) + 2 k + 8 = 7$]

#math.equation(block: true, alt: "− open parenthesis the fraction 1 over 4 a minus the fraction 3 over 4 close parenthesis plus the fraction 5 over 4 a equals −2")[$"−" ( frac(1, 4) a − frac(3, 4) ) + frac(5, 4) a = −2$]

#solutionbox[
#math.equation(block: true, alt: "a equals minus the fraction 11 over 4")[$a = − frac(11, 4)$]
]

#math.equation(block: true, alt: "− open parenthesis the fraction 2 over 3 d minus the fraction 1 over 3 close parenthesis plus the fraction 5 over 3 d equals −4")[$"−" ( frac(2, 3) d − frac(1, 3) ) + frac(5, 3) d = −4$]

#math.equation(block: true, alt: "8 open parenthesis 4 x plus 5 close parenthesis minus 5 open parenthesis 6 x close parenthesis minus x")[$8 ( 4 x + 5 ) − 5 ( 6 x ) − x$] #linebreak() #math.equation(block: true, alt: "equals 53 minus 6 open parenthesis x plus 1 close parenthesis plus 3 open parenthesis 2 x plus 2 close parenthesis")[$= 53 − 6 ( x + 1 ) + 3 ( 2 x + 2 )$]

#solutionbox[
#math.equation(block: true, alt: "x equals 13")[$x = 13$]
]

#math.equation(block: true, alt: "6 open parenthesis 9 y minus 1 close parenthesis minus 10 open parenthesis 5 y close parenthesis minus 3 y")[$6 ( 9 y − 1 ) − 10 ( 5 y ) − 3 y$] #linebreak() #math.equation(block: true, alt: "equals 22 minus 4 open parenthesis 2 y minus 12 close parenthesis plus 8 open parenthesis y minus 6 close parenthesis")[$= 22 − 4 ( 2 y − 12 ) + 8 ( y − 6 )$]

#strong[Translate to an Equation and Solve]

In the following exercises, translate to an equation and then solve it.

Nine more than #math.equation(block: false, alt: "x")[$x$] is equal to 52.

#solutionbox[
#math.equation(block: true, alt: "x plus 9 equals 52 ; x equals 43")[$x + 9 = 52 ; x = 43$]
]

The sum of #emph[x] and #math.equation(block: false, alt: "−15")[$−15$] is 23.

Ten less than #emph[m] is #math.equation(block: false, alt: "−14")[$−14$].

#solutionbox[
#math.equation(block: true, alt: "m minus 10 equals −14 ; m equals −4")[$m − 10 = −14 ; m = −4$]
]

Three less than #emph[y] is #math.equation(block: false, alt: "−19")[$−19$].

The sum of #emph[y] and #math.equation(block: false, alt: "−30")[$−30$] is 40.

#solutionbox[
#math.equation(block: true, alt: "y plus open parenthesis −30 close parenthesis equals 40 ; y equals 70")[$y + ( −30 ) = 40 ; y = 70$]
]

Twelve more than #emph[p] is equal to 67.

The difference of #math.equation(block: false, alt: "9 x and 8 x")[$9 x #h(0.2em) "and" #h(0.2em) 8 x$] is 107.

#solutionbox[
#math.equation(block: true, alt: "9 x minus 8 x equals 107 ; 107")[$9 x − 8 x = 107 ; 107$]
]

The difference of #math.equation(block: false, alt: "5 c and 4 c")[$5 c #h(0.2em) "and" #h(0.2em) 4 c$] is 602.

The difference of #math.equation(block: false, alt: "n")[$n$] and #math.equation(block: false, alt: "the fraction 1 over 6")[$frac(1, 6)$] is #math.equation(block: false, alt: "the fraction 1 over 2")[$frac(1, 2)$].

#solutionbox[
#math.equation(block: true, alt: "n minus the fraction 1 over 6 equals the fraction 1 over 2 ; the fraction 2 over 3")[$n − frac(1, 6) = frac(1, 2) ; frac(2, 3)$]
]

The difference of #math.equation(block: false, alt: "f")[$f$] and #math.equation(block: false, alt: "the fraction 1 over 3")[$frac(1, 3)$] is #math.equation(block: false, alt: "the fraction 1 over 12")[$frac(1, 12)$].

The sum of #math.equation(block: false, alt: "−4 n")[$−4 n$] and #math.equation(block: false, alt: "5 n")[$5 n$] is #math.equation(block: false, alt: "−82")[$−82$].

#solutionbox[
#math.equation(block: true, alt: "−4 n plus 5 n equals −82 ; minus 82")[$−4 n + 5 n = −82 ; − 82$]
]

The sum of #math.equation(block: false, alt: "−9 m")[$−9 m$] and #math.equation(block: false, alt: "10 m")[$10 m$] is #math.equation(block: false, alt: "−95")[$−95$].

#strong[Translate and Solve Applications]

In the following exercises, translate into an equation and solve.

#strong[Distance] Avril rode her bike a total of 18 miles, from home to the library and then to the beach. The distance from Avril’s house to the library is 7 miles. What is the distance from the library to the beach?

#solutionbox[
11 miles
]

#strong[Reading] Jeff read a total of 54 pages in his History and Sociology textbooks. He read 41 pages in his History textbook. How many pages did he read in his Sociology textbook?

#strong[Age] Eva’s daughter is 15 years younger than her son. Eva’s son is 22 years old. How old is her daughter?

#solutionbox[
7 years old
]

#strong[Age] Pablo’s father is 3 years older than his mother. Pablo’s mother is 42 years old. How old is his father?

#strong[Groceries] For a family birthday dinner, Celeste bought a turkey that weighed 5 pounds less than the one she bought for Thanksgiving. The birthday turkey weighed 16 pounds. How much did the Thanksgiving turkey weigh?

#solutionbox[
21 pounds
]

#strong[Weight] Allie weighs 8 pounds less than her twin sister Lorrie. Allie weighs 124 pounds. How much does Lorrie weigh?

#strong[Health] Connor’s temperature was 0.7 degrees higher this morning than it had been last night. His temperature this morning was 101.2 degrees. What was his temperature last night?

#solutionbox[
100.5 degrees
]

#strong[Health] The nurse reported that Tricia’s daughter had gained 4.2 pounds since her last checkup and now weighs 31.6 pounds. How much did Tricia’s daughter weigh at her last checkup?

#strong[Salary] Ron’s paycheck this week was \$17.43 less than his paycheck last week. His paycheck this week was \$103.76. How much was Ron’s paycheck last week?

#solutionbox[
\$121.19
]

#strong[Textbooks] Melissa’s math book cost \$22.85 less than her art book cost. Her math book cost \$93.75. How much did her art book cost?


==== Everyday Math

#strong[Construction] Miguel wants to drill a hole for a #math.equation(block: false, alt: "the fraction 5 over 8")[$frac(5, 8)$] inch screw. The hole should be #math.equation(block: false, alt: "the fraction 1 over 12")[$frac(1, 12)$] inch smaller than the screw. Let #math.equation(block: false, alt: "d")[$d$] equal the size of the hole he should drill. Solve the equation #math.equation(block: false, alt: "d plus the fraction 1 over 12 equals the fraction 5 over 8")[$d + frac(1, 12) = frac(5, 8)$] to see what size the hole should be.

#solutionbox[
#math.equation(block: true, alt: "d equals the fraction 13 over 24 inch")[$d = frac(13, 24) #h(0.2em) "inch"$]
]

#strong[Baking] Kelsey needs #math.equation(block: false, alt: "the fraction 2 over 3")[$frac(2, 3)$] cup of sugar for the cookie recipe she wants to make. She only has #math.equation(block: false, alt: "the fraction 3 over 8")[$frac(3, 8)$] cup of sugar and will borrow the rest from her neighbor. Let #math.equation(block: false, alt: "s")[$s$] equal the amount of sugar she will borrow. Solve the equation #math.equation(block: false, alt: "the fraction 3 over 8 plus s equals the fraction 2 over 3")[$frac(3, 8) + s = frac(2, 3)$] to find the amount of sugar she should ask to borrow.


==== Writing Exercises

Is #math.equation(block: false, alt: "−8")[$−8$] a solution to the equation #math.equation(block: false, alt: "3 x equals 16 minus 5 x")[$3 x = 16 − 5 x$]? How do you know?

#solutionbox[
No. Justifications will vary.
]

What is the first step in your solution to the equation #math.equation(block: false, alt: "10 x plus 2 equals 4 x plus 26")[$10 x + 2 = 4 x + 26$]?


==== Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

#figure(figph[This is a table that has six rows and four columns. In the first row, which is a header row, the cells read from left to right “I can…,” “Confidently,” “With some help,” and “No-I don’t get it!” The first column below “I can…” reads “verify a solution of an equation,” “solve equations using the subtraction and addition properties of equality,” “solve equations that require simplification,” “translate to an equation and solve,” and “translate and solve applications.” The rest of the cells are blank.], alt: "This is a table that has six rows and four columns. In the first row, which is a header row, the cells read from left to right “I can…,” “Confidently,” “With some help,” and “No-I don’t get it!” The first column below “I can…” reads “verify a solution of an equation,” “solve equations using the subtraction and addition properties of equality,” “solve equations that require simplification,” “translate to an equation and solve,” and “translate and solve applications.” The rest of the cells are blank.", caption: none)

ⓑ If most of your checks were:

#strong[…confidently.] Congratulations! You have achieved your goals in this section! Reflect on the study skills you used so that you can continue to use them. What did you do to become confident of your ability to do these things? Be specific!

#strong[…with some help.] This must be addressed quickly as topics you do not master become potholes in your road to success. Math is sequential - every topic builds upon previous work. It is important to make sure you have a strong foundation before you move on. Whom can you ask for help? Your fellow classmates and instructor are good resources. Is there a place on campus where math tutors are available? Can your study skills be improved?

#strong[…no - I don’t get it!] This is critical and you must not ignore it. You need to get help immediately or you will quickly be overwhelmed. See your instructor as soon as possible to discuss your situation. Together you can come up with a plan to get you the help you need.