#set document(title: "2.2 Evaluate, Simplify, and Translate Expressions", author: "OpenStax / XYZ Homework") #set page(width: 8.5in, height: auto, margin: 1in) #import "@preview/cetz:0.5.2" #set text(font: ("STIX Two Text", "Libertinus Serif", "New Computer Modern"), size: 10.5pt, lang: "en") #show math.equation: set text(font: ("STIX Two Math", "New Computer Modern Math")) #set par(justify: true, leading: 0.62em, spacing: 0.9em) #set enum(spacing: 1.1em) // room between list items so tall inline fractions don't collide #set list(spacing: 1.1em) #set table(stroke: 0.5pt + rgb("#c7ccd3")) #let BLUE = rgb("#183B6F") // brand navy — section bars + example/solution labels (white on navy 11.09:1) #let ORANGE = rgb("#A94509") // brand primary-700 — AA-safe deep orange for TEXT (5.93:1 on white; raw brand #F37021 is 2.94:1 and must never carry text) #let RED = rgb("#DC2626") // brand error-600 #let GREEN = rgb("#059669") // brand success-600 (decoration only; small green text uses green-text #007942) #show heading.where(level: 1): it => block(width: 100%, above: 0pt, below: 16pt, fill: gradient.linear(BLUE, rgb("#2C5AA0")), inset: (x: 14pt, y: 12pt), radius: 3pt, text(fill: white, weight: "bold", size: 19pt, it.body)) #show heading.where(level: 2): it => block(width: 100%, above: 18pt, below: 10pt, fill: BLUE, inset: (x: 10pt, y: 6pt), radius: 2pt, text(fill: white, weight: "bold", size: 12pt, it.body)) #show heading.where(level: 3): it => text(fill: ORANGE, weight: "bold", size: 12.5pt, it.body) #show heading.where(level: 4): it => text(fill: BLUE, weight: "bold", size: 10.5pt, it.body) #let examplebox(label, title, body) = block(width: 100%, breakable: true, fill: rgb("#EFF1F5"), stroke: 0.5pt + rgb("#CFDDF0"), radius: 4pt, inset: 10pt, above: 12pt, below: 12pt)[ #block(below: 6pt)[#box(fill: BLUE, inset: (x: 6pt, y: 2pt), radius: 2pt, text(fill: white, weight: "bold", size: 8.5pt, label)) #h(0.4em) #strong[#title]] #body] // rail = decorative left rule (raw brand token); labelcolor = AA-safe label text shade #let notebox(label, rail, labelcolor, tint, body) = block(width: 100%, breakable: true, fill: tint, stroke: (left: 3pt + rail), inset: (left: 10pt, rest: 8pt), radius: (right: 4pt), above: 11pt, below: 11pt)[ #text(fill: labelcolor, weight: "bold", size: 7.5pt, tracking: 0.5pt)[#upper(label)] #linebreak() #body] #let solutionbox(body) = block(above: 4pt, below: 8pt)[ #text(fill: BLUE, weight: "bold", size: 8.5pt)[Solution] #linebreak() #body] #let figph(msg) = block(width: 100%, height: 60pt, fill: rgb("#f6f7f9"), stroke: (paint: rgb("#c7ccd3"), dash: "dashed"), radius: 4pt, inset: 10pt)[ #align(center + horizon, text(fill: rgb("#889"), style: "italic", size: 9pt, msg))] // Standardize inlined figure sizes: measure the natural CeTZ canvas, then scale to a // consistent envelope (aspect-aware; see build_typst.py FIG_* constants). Unlike the // print preamble, dimensions are FLOORED: in an editor a user can trim a figure to a // degenerate 1-D shape (a bare line), and w/h or tw/w would then divide by zero. #let _STD_W = 3.5 #let _WIDE_W = 5.6 #let _MAX_H = 3.4 #let _ASPECT_WIDE = 2.2 #let _UPSCALE_MAX = 1.15 #let stdfig(body) = context { let m = measure(body) let w = calc.max(m.width / 1in, 0.01) let h = calc.max(m.height / 1in, 0.01) let tw = if w / h > _ASPECT_WIDE { _WIDE_W } else { _STD_W } let s = calc.min(tw / w, _MAX_H / h, _UPSCALE_MAX) align(center, box(scale(x: s * 100%, y: s * 100%, reflow: true, body))) } #show figure: set block(breakable: false) #set figure(gap: 8pt) #show figure.caption: set text(size: 8.5pt, fill: rgb("#555")) == 2.2#h(0.6em)Evaluate, Simplify, and Translate Expressions #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Before you get started, take this readiness quiz. Is #math.equation(block: false, alt: "n divided by 5")[$n ÷ 5$] an expression or an equation? #linebreak() If you missed this problem, review . #solutionbox[ expression ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Simplify #math.equation(block: false, alt: "4 to the power 5 .")[$4^(5) .$] #linebreak() If you missed this problem, review . #solutionbox[ #math.equation(block: true, alt: "1 , 024")[$1 , 024$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Simplify #math.equation(block: false, alt: "1 plus 8 times 9 .")[$1 + 8 ⋅ 9 .$] #linebreak() If you missed this problem, review . #solutionbox[ #math.equation(block: true, alt: "73")[$73$] ] ] === Evaluate Algebraic Expressions In the last section, we simplified expressions using the order of operations. In this section, we’ll evaluate expressions—again following the order of operations. To #strong[evaluate] an algebraic expression means to find the value of the expression when the #strong[variable] is replaced by a given number. To evaluate an #strong[expression], we substitute the given number for the variable in the expression and then simplify the expression using the order of operations. #examplebox("Example 1")[][ Evaluate #math.equation(block: false, alt: "x plus 7")[$x + 7$] when + ⓐ #math.equation(block: false, alt: "x equals 3")[$#h(0.2em) x = 3$] + ⓑ #math.equation(block: false, alt: "x equals 12")[$#h(0.2em) x = 12$] #solutionbox[ ⓐ To evaluate, substitute #math.equation(block: false, alt: "3")[$3$] for #math.equation(block: false, alt: "x")[$x$] in the expression, and then simplify. #linebreak() #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#figure(figph[The mathematical expression 'x + 7' is displayed in black text on a white background.], alt: "The mathematical expression 'x + 7' is displayed in black text on a white background.", caption: none)]), [Substitute.], [#figure(figph[The image displays a simple mathematical equation, '3 + 7', with the number 3 in red and the plus sign and number 7 in black, all against a white background.], alt: "The image displays a simple mathematical equation, '3 + 7', with the number 3 in red and the plus sign and number 7 in black, all against a white background.", caption: none)], [Add.], [#figure(figph[The number 10 is displayed.], alt: "The number 10 is displayed.", caption: none)], )) When #math.equation(block: false, alt: "x equals 3 ,")[$x = 3 ,$] the expression #math.equation(block: false, alt: "x plus 7")[$x + 7$] has a value of #math.equation(block: false, alt: "10 .")[$10 .$] ⓑ To evaluate, substitute #math.equation(block: false, alt: "12")[$12$] for #math.equation(block: false, alt: "x")[$x$] in the expression, and then simplify. #linebreak() #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#figure(figph[A mathematical expression consisting of the variable x, a plus sign, and the number 7, forming 'x + 7' in bold, sans-serif font on a white background.], alt: "A mathematical expression consisting of the variable x, a plus sign, and the number 7, forming 'x + 7' in bold, sans-serif font on a white background.", caption: none)]), [Substitute.], [#figure(figph[A mathematical expression showing the addition of two numbers, '12' in red and '7' in black, separated by a black plus sign: 12 + 7.], alt: "A mathematical expression showing the addition of two numbers, '12' in red and '7' in black, separated by a black plus sign: 12 + 7.", caption: none)], [Add.], [#figure(figph[The number 19 is displayed.], alt: "The number 19 is displayed.", caption: none)], )) When #math.equation(block: false, alt: "x equals 12 ,")[$x = 12 ,$] the expression #math.equation(block: false, alt: "x plus 7")[$x + 7$] has a value of #math.equation(block: false, alt: "19 .")[$19 .$] Notice that we got different results for parts ⓐ and ⓑ even though we started with the same expression. This is because the values used for #math.equation(block: false, alt: "x")[$x$] were different. When we evaluate an expression, the value varies depending on the value used for the variable. ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Evaluate: #math.equation(block: true, alt: "y plus 4 when")[$y + 4 #h(0.2em) "when" #h(0.2em)$] + ⓐ #math.equation(block: false, alt: "y equals 6")[$#h(0.2em) y = 6 #h(0.2em) #h(0.2em)$] + ⓑ #math.equation(block: false, alt: "y equals 15")[$#h(0.2em) y = 15$] #solutionbox[ + ⓐ 10 + ⓑ 19 ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Evaluate: #math.equation(block: true, alt: "a minus 5 when")[$a − 5 #h(0.2em) "when" #h(0.2em)$] + ⓐ #math.equation(block: false, alt: "a equals 9")[$#h(0.2em) a = 9 #h(0.2em) #h(0.2em)$] + ⓑ #math.equation(block: false, alt: "a equals 17")[$#h(0.2em) a = 17$] #solutionbox[ + ⓐ 4 + ⓑ 12 ] ] #examplebox("Example 2")[][ Evaluate #math.equation(block: false, alt: "9 x minus 2 , when")[$9 x − 2 , "when" #h(0.2em)$] + ⓐ #math.equation(block: false, alt: "x equals 5")[$#h(0.2em) x = 5 #h(0.2em) #h(0.2em)$] + ⓑ #math.equation(block: false, alt: "x equals 1")[$#h(0.2em) x = 1$] #solutionbox[ Remember #math.equation(block: false, alt: "a b")[$a b$] means #math.equation(block: false, alt: "a")[$a$] times #math.equation(block: false, alt: "b ,")[$b ,$] so #math.equation(block: false, alt: "9 x")[$9 x$] means #math.equation(block: false, alt: "9")[$9$] times #math.equation(block: false, alt: "x .")[$x .$] ⓐ To evaluate the expression when #math.equation(block: false, alt: "x equals 5 ,")[$x = 5 ,$] we substitute #math.equation(block: false, alt: "5")[$5$] for #math.equation(block: false, alt: "x ,")[$x ,$] and then simplify. #linebreak() #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#figure(figph[A mathematical expression '9x - 2' is displayed in a clear, digital font against a white background.], alt: "A mathematical expression '9x - 2' is displayed in a clear, digital font against a white background.", caption: none)]), [#figure(figph[The text 'Substitute 5 for x.' is shown, with the number 5 highlighted in red and the rest of the text in a dark teal color against a white background.], alt: "The text 'Substitute 5 for x.' is shown, with the number 5 highlighted in red and the rest of the text in a dark teal color against a white background.", caption: none)], [#figure(figph[A mathematical expression '9.5-2' is displayed. The number '5' in '9.5' is rendered in red, while the other numbers '9' and '2', as well as the dot and minus sign, are in black.], alt: "A mathematical expression '9.5-2' is displayed. The number '5' in '9.5' is rendered in red, while the other numbers '9' and '2', as well as the dot and minus sign, are in black.", caption: none)], [Multiply.], [#figure(figph[The image displays the mathematical expression '45 - 2' in a simple black font on a white background, indicating a subtraction problem.], alt: "The image displays the mathematical expression '45 - 2' in a simple black font on a white background, indicating a subtraction problem.", caption: none)], [Subtract.], [#figure(figph[The number 43 is displayed in black text on a plain white background.], alt: "The number 43 is displayed in black text on a plain white background.", caption: none)], )) ⓑ To evaluate the expression when #math.equation(block: false, alt: "x equals 1 ,")[$x = 1 ,$] we substitute #math.equation(block: false, alt: "1")[$1$] for #math.equation(block: false, alt: "x ,")[$x ,$] and then simplify. #linebreak() #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#figure(figph[The mathematical expression '9x - 2' is displayed in a clear, dark font against a white background.], alt: "The mathematical expression '9x - 2' is displayed in a clear, dark font against a white background.", caption: none)]), [#figure(figph[The text 'Substitute 1 for x.' is shown, with the number 1 highlighted in red, on a plain white background.], alt: "The text 'Substitute 1 for x.' is shown, with the number 1 highlighted in red, on a plain white background.", caption: none)], [#figure(figph[The image shows a mathematical expression: 9(1) - 2. The number 1 is highlighted in red, indicating it might be a substituted value or a point of focus in the calculation.], alt: "The image shows a mathematical expression: 9(1) - 2. The number 1 is highlighted in red, indicating it might be a substituted value or a point of focus in the calculation.", caption: none)], [Multiply.], [#figure(figph[The mathematical expression '9 - 2' is displayed in black text on a white background, representing a subtraction problem.], alt: "The mathematical expression '9 - 2' is displayed in black text on a white background, representing a subtraction problem.", caption: none)], [Subtract.], [#figure(figph[A number 7 is visible in the bottom right corner of a plain white background.], alt: "A number 7 is visible in the bottom right corner of a plain white background.", caption: none)], )) Notice that in part ⓐ that we wrote #math.equation(block: false, alt: "9 times 5")[$9 ⋅ 5$] and in part ⓑ we wrote #math.equation(block: false, alt: "9 open parenthesis 1 close parenthesis .")[$9 ( 1 ) .$] Both the dot and the parentheses tell us to multiply. ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Evaluate: #math.equation(block: true, alt: "8 x minus 3 , when")[$8 x − 3 , "when" #h(0.2em)$] + ⓐ #math.equation(block: false, alt: "x equals 2")[$#h(0.2em) x = 2 #h(0.2em) #h(0.2em)$] + ⓑ #math.equation(block: false, alt: "x equals 1")[$#h(0.2em) x = 1$] #solutionbox[ + ⓐ 13 + ⓑ 5 ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Evaluate: #math.equation(block: true, alt: "4 y minus 4 , when")[$4 y − 4 , "when" #h(0.2em)$] + ⓐ #math.equation(block: false, alt: "y equals 3")[$#h(0.2em) y = 3 #h(0.2em) #h(0.2em)$] + ⓑ #math.equation(block: false, alt: "y equals 5")[$#h(0.2em) y = 5$] #solutionbox[ + ⓐ 8 + ⓑ 16 ] ] #examplebox("Example 3")[][ Evaluate #math.equation(block: false, alt: "x squared")[$x^(2)$] when #math.equation(block: false, alt: "x equals 10 .")[$x = 10 .$] #solutionbox[ We substitute #math.equation(block: false, alt: "10")[$10$] for #math.equation(block: false, alt: "x ,")[$x ,$] and then simplify the expression. #linebreak() #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#figure(figph[The image displays the mathematical expression 'x^2' with the letter 'x' in lowercase and the number '2' as a superscript, indicating 'x squared'.], alt: "The image displays the mathematical expression 'x^2' with the letter 'x' in lowercase and the number '2' as a superscript, indicating 'x squared'.", caption: none)]), [#figure(figph[The image displays text that instructs the user to 'Substitute 10 for x.' The number 10 is highlighted in red, while the rest of the text is in a dark teal color.], alt: "The image displays text that instructs the user to 'Substitute 10 for x.' The number 10 is highlighted in red, while the rest of the text is in a dark teal color.", caption: none)], [#figure(figph[The number 10 is shown in red with a black superscript 2, representing 10 squared or 10 to the power of 2.], alt: "The number 10 is shown in red with a black superscript 2, representing 10 squared or 10 to the power of 2.", caption: none)], [Use the definition of exponent.], [#figure(figph[A mathematical expression displays '10 . 10', indicating the multiplication of ten by ten, which equals one hundred. The numbers are bold and centered against a white background.], alt: "A mathematical expression displays '10 . 10', indicating the multiplication of ten by ten, which equals one hundred. The numbers are bold and centered against a white background.", caption: none)], [Multiply.], [#figure(figph[The number 100 is displayed in bold black text on a white background.], alt: "The number 100 is displayed in bold black text on a white background.", caption: none)], )) When #math.equation(block: false, alt: "x equals 10 ,")[$x = 10 ,$] the expression #math.equation(block: false, alt: "x squared")[$x^(2)$] has a value of #math.equation(block: false, alt: "100 .")[$100 .$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Evaluate: #math.equation(block: true, alt: "x squared when x equals 8 .")[$x^(2) #h(0.2em) "when" #h(0.2em) x = 8 .$] #solutionbox[ 64 ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Evaluate: #math.equation(block: true, alt: "x cubed when x equals 6 .")[$x^(3) #h(0.2em) "when" #h(0.2em) x = 6 .$] #solutionbox[ 216 ] ] #examplebox("Example 4")[][ #math.equation(block: true, alt: "Evaluate 2 to the power x when x equals 5 .")[$"Evaluate" #h(0.2em) 2^(x) #h(0.2em) "when" #h(0.2em) x = 5 .$] #solutionbox[ In this expression, the variable is an exponent. #linebreak() #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#figure(figph[The expression '2x' is shown in black text against a white background.], alt: "The expression '2x' is shown in black text against a white background.", caption: none)]), [#figure(figph[The text 'Substitute 5 for x.' is shown in a dark teal font on a white background, with the number 5 highlighted in red.], alt: "The text 'Substitute 5 for x.' is shown in a dark teal font on a white background, with the number 5 highlighted in red.", caption: none)], [#figure(figph[The mathematical expression 2^5 is shown with the number 2 in black and the exponent 5 in a distinct red color, indicating 2 raised to the power of 5.], alt: "The mathematical expression 2^5 is shown with the number 2 in black and the exponent 5 in a distinct red color, indicating 2 raised to the power of 5.", caption: none)], [Use the definition of exponent.], [#figure(figph[The mathematical expression showing the number 2 multiplied by itself five times, represented as 2 . 2 . 2 . 2 . 2, which is equal to 32 or 2^5.], alt: "The mathematical expression showing the number 2 multiplied by itself five times, represented as 2 . 2 . 2 . 2 . 2, which is equal to 32 or 2^5.", caption: none)], [Multiply.], [#figure(figph[The number 32 is displayed.], alt: "The number 32 is displayed.", caption: none)], )) When #math.equation(block: false, alt: "x equals 5 ,")[$x = 5 ,$] the expression #math.equation(block: false, alt: "2 to the power x")[$2^(x)$] has a value of #math.equation(block: false, alt: "32 .")[$32 .$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Evaluate: #math.equation(block: true, alt: "2 to the power x when x equals 6 .")[$2^(x) #h(0.2em) "when" #h(0.2em) x = 6 .$] #solutionbox[ 64 ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Evaluate: #math.equation(block: true, alt: "3 to the power x when x equals 4 .")[$3^(x) #h(0.2em) "when" #h(0.2em) x = 4 .$] #solutionbox[ 81 ] ] #examplebox("Example 5")[][ #math.equation(block: true, alt: "Evaluate 3 x plus 4 y minus 6 when x equals 10 and y equals 2 .")[$"Evaluate" #h(0.2em) 3 x + 4 y − 6 #h(0.2em) "when" #h(0.2em) x = 10 #h(0.2em) "and" #h(0.2em) y = 2 .$] #solutionbox[ This expression contains two variables, so we must make two substitutions. #linebreak() #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#figure(figph[The mathematical expression 3x + 4y - 6 is displayed on a white background.], alt: "The mathematical expression 3x + 4y - 6 is displayed on a white background.", caption: none)]), [#figure(figph[The image shows the text 'Substitute 10 for x and 2 for y.'], alt: "The image shows the text 'Substitute 10 for x and 2 for y.'", caption: none)], [#figure(figph[A mathematical expression reads as three times ten, plus four times two, minus six. The '10' is highlighted in red, and the '2' is highlighted in light blue.], alt: "A mathematical expression reads as three times ten, plus four times two, minus six. The '10' is highlighted in red, and the '2' is highlighted in light blue.", caption: none)], [Multiply.], [#figure(figph[A mathematical expression featuring the numbers 30, 8, and 6, connected by a plus sign and a minus sign: 30 + 8 - 6.], alt: "A mathematical expression featuring the numbers 30, 8, and 6, connected by a plus sign and a minus sign: 30 + 8 - 6.", caption: none)], [Add and subtract left to right.], [#figure(figph[The number 32 in black text on a plain white background.], alt: "The number 32 in black text on a plain white background.", caption: none)], )) When #math.equation(block: false, alt: "x equals 10")[$x = 10$] and #math.equation(block: false, alt: "y equals 2 ,")[$y = 2 ,$] the expression #math.equation(block: false, alt: "3 x plus 4 y minus 6")[$3 x + 4 y − 6$] has a value of #math.equation(block: false, alt: "32 .")[$32 .$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Evaluate: #math.equation(block: true, alt: "2 x plus 5 y minus 4 when x equals 11 and y equals 3")[$2 x + 5 y − 4 #h(0.2em) "when" #h(0.2em) x = 11 #h(0.2em) "and" #h(0.2em) y = 3$] #solutionbox[ 33 ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Evaluate: #math.equation(block: true, alt: "5 x minus 2 y minus 9 when x equals 7 and y equals 8")[$5 x − 2 y − 9 #h(0.2em) "when" #h(0.2em) x = 7 #h(0.2em) "and" #h(0.2em) y = 8$] #solutionbox[ 10 ] ] #examplebox("Example 6")[][ #math.equation(block: true, alt: "Evaluate 2 x squared plus 3 x plus 8 when x equals 4 .")[$"Evaluate" #h(0.2em) 2 x^(2) + 3 x + 8 #h(0.2em) "when" #h(0.2em) x = 4 .$] #solutionbox[ We need to be careful when an expression has a variable with an exponent. In this expression, #math.equation(block: false, alt: "2 x squared")[$2 x^(2)$] means #math.equation(block: false, alt: "2 times x times x")[$2 ⋅ x ⋅ x$] and is different from the expression #math.equation(block: false, alt: "open parenthesis 2 x close parenthesis squared ,")[$attach(( 2 x ), t: 2) ,$] which means #math.equation(block: false, alt: "2 x times 2 x .")[$2 x ⋅ 2 x .$] #linebreak() #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#figure(figph[The mathematical expression '2x^2 + 3x + 8' is displayed in a clear, dark gray font against a plain white background, appearing as a standard quadratic equation or polynomial.], alt: "The mathematical expression '2x^2 + 3x + 8' is displayed in a clear, dark gray font against a plain white background, appearing as a standard quadratic equation or polynomial.", caption: none)]), [#figure(figph[The image shows text that says 'Substitute 4 for each x.' The number 4 is highlighted in red, while the rest of the text is in a dark blue-grey color.], alt: "The image shows text that says 'Substitute 4 for each x.' The number 4 is highlighted in red, while the rest of the text is in a dark blue-grey color.", caption: none)], [#figure(figph[The image shows the numerical evaluation of the expression 2x^2 + 3x + 8 where x is replaced by 4. The number 4 is highlighted in red in both occurrences.], alt: "The image shows the numerical evaluation of the expression 2x^2 + 3x + 8 where x is replaced by 4. The number 4 is highlighted in red in both occurrences.", caption: none)], [Simplify #math.equation(block: false, alt: "4 squared")[$4^(2)$].], [#figure(figph[A mathematical expression showing the sum of products and a single number: 2 multiplied by 16, plus 3 multiplied by 4, plus 8.], alt: "A mathematical expression showing the sum of products and a single number: 2 multiplied by 16, plus 3 multiplied by 4, plus 8.", caption: none)], [Multiply.], [#figure(figph[The image displays the mathematical expression '32 + 12 + 8' in a dark font against a white background. Each number and operator is clearly visible, forming a simple addition problem.], alt: "The image displays the mathematical expression '32 + 12 + 8' in a dark font against a white background. Each number and operator is clearly visible, forming a simple addition problem.", caption: none)], [Add.], [#figure(figph[The number 52 is displayed in a black, sans-serif font on a white background.], alt: "The number 52 is displayed in a black, sans-serif font on a white background.", caption: none)], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Evaluate: #math.equation(block: true, alt: "3 x squared plus 4 x plus 1 when x equals 3 .")[$3 x^(2) + 4 x + 1 #h(0.2em) "when" #h(0.2em) x = 3 .$] #solutionbox[ 40 ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Evaluate: #math.equation(block: true, alt: "6 x squared minus 4 x minus 7 when x equals 2 .")[$6 x^(2) − 4 x − 7 #h(0.2em) "when" #h(0.2em) x = 2 .$] #solutionbox[ 9 ] ] === Identify Terms, Coefficients, and Like Terms Algebraic expressions are made up of #emph[terms]. A #strong[term] is a constant or the product of a constant and one or more variables. Some examples of terms are #math.equation(block: false, alt: "7 , y , 5 x squared , 9 a , and 13 x y .")[$7 , y , 5 x^(2) , 9 a , "and" #h(0.2em) 13 x y .$] The constant that multiplies the variable(s) in a term is called the #strong[coefficient]. We can think of the coefficient as the number #emph[in front of] the variable. The coefficient of the term #math.equation(block: false, alt: "3 x")[$3 x$] is #math.equation(block: false, alt: "3 .")[$3 .$] When we write #math.equation(block: false, alt: "x ,")[$x ,$] the coefficient is #math.equation(block: false, alt: "1 ,")[$1 ,$] since #math.equation(block: false, alt: "x equals 1 times x .")[$x = 1 ⋅ x .$] gives the coefficients for each of the terms in the left column. #figure(table( columns: 2, align: left, inset: 6pt, table.header([Term], [Coefficient]), [#math.equation(block: false, alt: "9 a")[$9 a$]], [#math.equation(block: false, alt: "9")[$9$]], [#math.equation(block: false, alt: "y")[$y$]], [#math.equation(block: false, alt: "1")[$1$]], [#math.equation(block: false, alt: "5 x squared")[$5 x^(2)$]], [#math.equation(block: false, alt: "5")[$5$]], )) An algebraic expression may consist of one or more terms added or subtracted. In this chapter, we will only work with terms that are added together. gives some examples of algebraic expressions with various numbers of terms. Notice that we include the operation before a term with it. #figure(table( columns: 2, align: left, inset: 6pt, table.header([Expression], [Terms]), [#math.equation(block: false, alt: "7")[$7$]], [#math.equation(block: false, alt: "7")[$7$]], [#math.equation(block: false, alt: "y")[$y$]], [#math.equation(block: false, alt: "y")[$y$]], [#math.equation(block: false, alt: "x plus 7")[$x + 7$]], [#math.equation(block: false, alt: "x , 7")[$x , 7$]], [#math.equation(block: false, alt: "2 x plus 7 y plus 4")[$2 x + 7 y + 4$]], [#math.equation(block: false, alt: "2 x , 7 y , 4")[$2 x , 7 y , 4$]], [#math.equation(block: false, alt: "3 x squared plus 4 x squared plus 5 y plus 3")[$3 x^(2) + 4 x^(2) + 5 y + 3$]], [#math.equation(block: false, alt: "3 x squared , 4 x squared , 5 y , 3")[$3 x^(2) , 4 x^(2) , 5 y , 3$]], )) #examplebox("Example 7")[][ Identify each term in the expression #math.equation(block: false, alt: "9 b plus 15 x squared plus a plus 6 .")[$9 b + 15 x^(2) + a + 6 .$] Then identify the coefficient of each term. #solutionbox[ The expression has four terms. They are #math.equation(block: false, alt: "9 b , 15 x squared , a ,")[$9 b , 15 x^(2) , a ,$] and #math.equation(block: false, alt: "6 .")[$6 .$] The coefficient of #math.equation(block: false, alt: "9 b")[$9 b$] is #math.equation(block: false, alt: "9 .")[$9 .$] The coefficient of #math.equation(block: false, alt: "15 x squared")[$15 x^(2)$] is #math.equation(block: false, alt: "15 .")[$15 .$] Remember that if no number is written before a variable, the coefficient is #math.equation(block: false, alt: "1 .")[$1 .$] So the coefficient of #math.equation(block: false, alt: "a")[$a$] is #math.equation(block: false, alt: "1 .")[$1 .$] The coefficient of a constant is the constant, so the coefficient of #math.equation(block: false, alt: "6")[$6$] is #math.equation(block: false, alt: "6 .")[$6 .$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Identify all terms in the given expression, and their coefficients: #math.equation(block: true, alt: "4 x plus 3 b plus 2")[$4 x + 3 b + 2$] #solutionbox[ The terms are 4#emph[x], 3#emph[b], and 2. The coefficients are 4, 3, and 2. ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Identify all terms in the given expression, and their coefficients: #math.equation(block: true, alt: "9 a plus 13 a squared plus a cubed")[$9 a + 13 a^(2) + a^(3)$] #solutionbox[ The terms are 9#emph[a], 13#emph[a]#super[2], and #emph[a]#super[3], The coefficients are 9, 13, and 1. ] ] Some terms share common traits. Look at the following terms. Which ones seem to have traits in common? #math.equation(block: true, alt: "5 x , 7 , n squared , 4 , 3 x , 9 n squared")[$5 x , 7 , n^(2) , 4 , 3 x , 9 n^(2)$]Which of these terms are like terms? - The terms #math.equation(block: false, alt: "7")[$7$] and #math.equation(block: false, alt: "4")[$4$] are both constant terms. - The terms #math.equation(block: false, alt: "5 x")[$5 x$] and #math.equation(block: false, alt: "3 x")[$3 x$] are both terms with #math.equation(block: false, alt: "x .")[$x .$] - The terms #math.equation(block: false, alt: "n squared")[$n^(2)$] and #math.equation(block: false, alt: "9 n squared")[$9 n^(2)$] both have #math.equation(block: false, alt: "n squared .")[$n^(2) .$] Terms are called #strong[like terms] if they have the same variables and exponents. All constant terms are also like terms. So among the terms #math.equation(block: false, alt: "5 x , 7 , n squared , 4 , 3 x , 9 n squared ,")[$5 x , 7 , n^(2) , 4 , 3 x , 9 n^(2) ,$] #math.equation(block: true, alt: "7 and 4 are like terms.")[$7 #h(0.2em) "and" #h(0.2em) 4 #h(0.2em) "are like terms."$]#math.equation(block: true, alt: "5 x and 3 x are like terms.")[$5 x #h(0.2em) "and" #h(0.2em) 3 x #h(0.2em) "are like terms."$]#math.equation(block: true, alt: "n squared and 9 n squared are like terms.")[$n^(2) #h(0.2em) "and" #h(0.2em) 9 n^(2) #h(0.2em) "are like terms."$]#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Like Terms] Terms that are either constants or have the same variables with the same exponents are like terms. ] #examplebox("Example 8")[][ Identify the like terms: + ⓐ #math.equation(block: false, alt: "y cubed , 7 x squared , 14 , 23 , 4 y cubed , 9 x , 5 x squared")[$#h(0.2em) y^(3) , 7 x^(2) , 14 , 23 , 4 y^(3) , 9 x , 5 x^(2)$] + ⓑ #math.equation(block: false, alt: "4 x squared plus 2 x plus 5 x squared plus 6 x plus 40 x plus 8 x y")[$#h(0.2em) 4 x^(2) + 2 x + 5 x^(2) + 6 x + 40 x + 8 x y$] #solutionbox[ ⓐ #math.equation(block: false, alt: "y cubed , 7 x squared , 14 , 23 , 4 y cubed , 9 x , 5 x squared")[$#h(0.2em) y^(3) , 7 x^(2) , 14 , 23 , 4 y^(3) , 9 x , 5 x^(2)$] Look at the variables and exponents. The expression contains #math.equation(block: false, alt: "y cubed , x squared , x ,")[$y^(3) , x^(2) , x ,$] and constants. The terms #math.equation(block: false, alt: "y cubed")[$y^(3)$] and #math.equation(block: false, alt: "4 y cubed")[$4 y^(3)$] are like terms because they both have #math.equation(block: false, alt: "y cubed .")[$y^(3) .$] The terms #math.equation(block: false, alt: "7 x squared")[$7 x^(2)$] and #math.equation(block: false, alt: "5 x squared")[$5 x^(2)$] are like terms because they both have #math.equation(block: false, alt: "x squared .")[$x^(2) .$] The terms #math.equation(block: false, alt: "14")[$14$] and #math.equation(block: false, alt: "23")[$23$] are like terms because they are both constants. The term #math.equation(block: false, alt: "9 x")[$9 x$] does not have any like terms in this list since no other terms have the variable #math.equation(block: false, alt: "x")[$x$] raised to the power of #math.equation(block: false, alt: "1 .")[$1 .$] ⓑ #math.equation(block: false, alt: "4 x squared plus 2 x plus 5 x squared plus 6 x plus 40 x plus 8 x y")[$#h(0.2em) 4 x^(2) + 2 x + 5 x^(2) + 6 x + 40 x + 8 x y$] Look at the variables and exponents. The expression contains the terms #math.equation(block: false, alt: "4 x squared , 2 x , 5 x squared , 6 x , 40 x , and 8 x y")[$4 x^(2) , 2 x , 5 x^(2) , 6 x , 40 x , "and" #h(0.2em) 8 x y$] The terms #math.equation(block: false, alt: "4 x squared")[$4 x^(2)$] and #math.equation(block: false, alt: "5 x squared")[$5 x^(2)$] are like terms because they both have #math.equation(block: false, alt: "x squared .")[$x^(2) .$] The terms #math.equation(block: false, alt: "2 x , 6 x , and 40 x")[$2 x , 6 x , "and" #h(0.2em) 40 x$] are like terms because they all have #math.equation(block: false, alt: "x .")[$x .$] The term #math.equation(block: false, alt: "8 x y")[$8 x y$] has no like terms in the given expression because no other terms contain the two variables #math.equation(block: false, alt: "x y .")[$x y .$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Identify the like terms in the list or the expression: #math.equation(block: true, alt: "9 , 2 x cubed , y squared , 8 x cubed , 15 , 9 y , 11 y squared")[$9 , 2 x^(3) , y^(2) , 8 x^(3) , 15 , 9 y , 11 y^(2)$] #solutionbox[ 9 and 15; 2#emph[x]#super[3] and 8#emph[x]#super[3]; #emph[y]#super[2] and 11#emph[y]#super[2] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Identify the like terms in the list or the expression: #math.equation(block: true, alt: "4 x cubed plus 8 x squared plus 19 plus 3 x squared plus 24 plus 6 x cubed")[$4 x^(3) + 8 x^(2) + 19 + 3 x^(2) + 24 + 6 x^(3)$] #solutionbox[ 4#emph[x]#super[3] and 6#emph[x]#super[3]; 8#emph[x]#super[2] and 3#emph[x]#super[2]; 19 and 24 ] ] === Simplify Expressions by Combining Like Terms We can simplify an expression by combining the #strong[like terms]. What do you think #math.equation(block: false, alt: "3 x plus 6 x")[$3 x + 6 x$] would simplify to? If you thought #math.equation(block: false, alt: "9 x ,")[$9 x ,$] you would be right! We can see why this works by writing both terms as addition problems. #figure(figph[The image shows the expression 3 x plus 6 x. The 3 x represents x plus x plus x. The 6 x represents x plus x plus x plus x plus x plus x. The expression 3 x plus 6 x becomes x plus x plus x plus x plus x plus x plus x plus x plus x. This simplifies to a total of 9 x's or the term 9 x.], alt: "The image shows the expression 3 x plus 6 x. The 3 x represents x plus x plus x. The 6 x represents x plus x plus x plus x plus x plus x. The expression 3 x plus 6 x becomes x plus x plus x plus x plus x plus x plus x plus x plus x. This simplifies to a total of 9 x's or the term 9 x.", caption: none) Add the #strong[coefficients] and keep the same variable. It doesn’t matter what #math.equation(block: false, alt: "x")[$x$] is. If you have #math.equation(block: false, alt: "3")[$3$] of something and add #math.equation(block: false, alt: "6")[$6$] more of the same thing, the result is #math.equation(block: false, alt: "9")[$9$] of them. For example, #math.equation(block: false, alt: "3")[$3$] oranges plus #math.equation(block: false, alt: "6")[$6$] oranges is #math.equation(block: false, alt: "9")[$9$] oranges. We will discuss the mathematical properties behind this later. The expression #math.equation(block: false, alt: "3 x plus 6 x")[$3 x + 6 x$] has only two terms. When an expression contains more terms, it may be helpful to rearrange the terms so that like terms are together. The #strong[Commutative Property of Addition] says that we can change the order of addends without changing the sum. So we could rearrange the following expression before combining like terms. #figure(figph[The image shows the expression 3 x plus 4 y plus 2 x plus 6 y. The position of the middle terms, 4 y and 2 x, can be switched so that the expression becomes 3 x plus 2 x plus 4 y plus 6 y. Now the terms containing x are together and the terms containing y are together.], alt: "The image shows the expression 3 x plus 4 y plus 2 x plus 6 y. The position of the middle terms, 4 y and 2 x, can be switched so that the expression becomes 3 x plus 2 x plus 4 y plus 6 y. Now the terms containing x are together and the terms containing y are together.", caption: none) Now it is easier to see the like terms to be combined. #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Combine like terms.] + Identify like terms. + Rearrange the expression so like terms are together. + Add the coefficients of the like terms. ] #examplebox("Example 9")[][ Simplify the expression: #math.equation(block: false, alt: "3 x plus 7 plus 4 x plus 5 .")[$3 x + 7 + 4 x + 5 .$] #solutionbox[ #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#figure(figph[A mathematical expression showing the sum of terms: three times x, plus seven, plus four times x, plus five (3x + 7 + 4x + 5). The numbers and the variable 'x' are displayed in black against a white background.], alt: "A mathematical expression showing the sum of terms: three times x, plus seven, plus four times x, plus five (3x + 7 + 4x + 5). The numbers and the variable 'x' are displayed in black against a white background.", caption: none)]), [Identify the like terms.], [#figure(figph[An algebraic expression is shown as 3x + 7 + 4x + 5. The terms 3x and 4x are in red, while 7 and 5 are in light blue, separated by black plus signs.], alt: "An algebraic expression is shown as 3x + 7 + 4x + 5. The terms 3x and 4x are in red, while 7 and 5 are in light blue, separated by black plus signs.", caption: none)], [Rearrange the expression, so the like terms are together.], [#figure(figph[A mathematical expression displaying the sum of two variable terms (3x and 4x) and two constant terms (7 and 5), written as 3x + 4x + 7 + 5.], alt: "A mathematical expression displaying the sum of two variable terms (3x and 4x) and two constant terms (7 and 5), written as 3x + 4x + 7 + 5.", caption: none)], [Add the coefficients of the like terms.], [#figure(figph[An algebraic expression being simplified by combining like terms: 3x + 4x becomes 7x, and 7 + 5 becomes 12. The final simplified form is 7x + 12.], alt: "An algebraic expression being simplified by combining like terms: 3x + 4x becomes 7x, and 7 + 5 becomes 12. The final simplified form is 7x + 12.", caption: none)], [The original expression is simplified to...], [#figure(figph[The image displays the mathematical expression '7x + 12' in a clear, dark font on a white background, representing a linear algebraic binomial.], alt: "The image displays the mathematical expression '7x + 12' in a clear, dark font on a white background, representing a linear algebraic binomial.", caption: none)], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Simplify: #math.equation(block: true, alt: "7 x plus 9 plus 9 x plus 8")[$7 x + 9 + 9 x + 8$] #solutionbox[ 16#emph[x] + 17 ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Simplify: #math.equation(block: true, alt: "5 y plus 2 plus 8 y plus 4 y plus 5")[$5 y + 2 + 8 y + 4 y + 5$] #solutionbox[ 17#emph[y] + 7 ] ] When any of the terms have negative coefficients, the procedure is the same, except that you have to subtract instead of adding to combine like terms. #examplebox("Example 10")[][ Simplify the expression: #math.equation(block: false, alt: "7 x squared plus 8 x minus x squared minus 4 x .")[$7 x^(2) + 8 x − x^(2) − 4 x .$] #solutionbox[ #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#figure(figph[The image shows the mathematical expression 7x^2 + 8x - x^2 - 4x. This expression represents a polynomial with terms involving x squared and x.], alt: "The image shows the mathematical expression 7x^2 + 8x - x^2 - 4x. This expression represents a polynomial with terms involving x squared and x.", caption: none)]), [Identify the like terms.], [#figure(figph[A mathematical expression: 7x^2 + 8x - x^2 - 4x. Terms with x^2 are red, and terms with x are blue. The expression simplifies to 6x^2 + 4x.], alt: "A mathematical expression: 7x^2 + 8x - x^2 - 4x. Terms with x^2 are red, and terms with x are blue. The expression simplifies to 6x^2 + 4x.", caption: none)], [Rearrange the expression so like terms are together.], [#figure(figph[A mathematical expression displaying algebraic terms: 7x^2 - x^2 + 8x - 4x. The terms are color-coded, with x^2 terms in red and x terms in blue, to highlight like terms for simplification.], alt: "A mathematical expression displaying algebraic terms: 7x^2 - x^2 + 8x - 4x. The terms are color-coded, with x^2 terms in red and x terms in blue, to highlight like terms for simplification.", caption: none)], [Add the coefficients of the like terms.], [#figure(figph[The image shows the mathematical expression 6x^2 + 4x, where 6x^2 is colored red and 4x is colored blue, connected by a black plus sign.], alt: "The image shows the mathematical expression 6x^2 + 4x, where 6x^2 is colored red and 4x is colored blue, connected by a black plus sign.", caption: none)], )) These are not like terms and cannot be combined. So #math.equation(block: false, alt: "6 x squared plus 4 x")[$6 x^(2) + 4 x$] is in simplest form. ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Simplify: #math.equation(block: true, alt: "3 x squared plus 9 x plus x squared plus 5 x")[$3 x^(2) + 9 x + x^(2) + 5 x$] #solutionbox[ 4#emph[x]#super[2] + 14#emph[x] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Simplify: #math.equation(block: true, alt: "11 y squared plus 8 y plus y squared plus 7 y")[$11 y^(2) + 8 y + y^(2) + 7 y$] #solutionbox[ 12#emph[y]#super[2] + 15#emph[y] ] ] === Translate Words to Algebraic Expressions In the previous section, we listed many operation symbols that are used in algebra, and then we translated expressions and equations into word phrases and sentences. Now we’ll reverse the process and translate word phrases into algebraic expressions. The symbols and variables we’ve talked about will help us do that. They are summarized in . #figure(table( columns: 3, align: left, inset: 6pt, table.header([Operation], [Phrase], [Expression]), [#strong[Addition]], [#math.equation(block: false, alt: "a")[$a$] plus #math.equation(block: false, alt: "b")[$b$] #linebreak() the sum of #math.equation(block: false, alt: "a")[$a$] and #math.equation(block: false, alt: "b")[$b$] #linebreak() #math.equation(block: false, alt: "a")[$a$] increased by #math.equation(block: false, alt: "b")[$b$] #linebreak() #math.equation(block: false, alt: "b")[$b$] more than #math.equation(block: false, alt: "a")[$a$] #linebreak() the total of #math.equation(block: false, alt: "a")[$a$] and #math.equation(block: false, alt: "b")[$b$] #linebreak() #math.equation(block: false, alt: "b")[$b$] added to #math.equation(block: false, alt: "a")[$a$]], [#math.equation(block: false, alt: "a plus b")[$a + b$]], [#strong[Subtraction]], [#math.equation(block: false, alt: "a")[$a$] minus #math.equation(block: false, alt: "b")[$b$] #linebreak() the difference of #math.equation(block: false, alt: "a")[$a$] and #math.equation(block: false, alt: "b")[$b$] #linebreak() #math.equation(block: false, alt: "b")[$b$] subtracted from #math.equation(block: false, alt: "a")[$a$] #linebreak() #math.equation(block: false, alt: "a")[$a$] decreased by #math.equation(block: false, alt: "b")[$b$] #linebreak() #math.equation(block: false, alt: "b")[$b$] less than #math.equation(block: false, alt: "a")[$a$]], [#math.equation(block: false, alt: "a minus b")[$a − b$]], [#strong[Multiplication]], [#math.equation(block: false, alt: "a")[$a$] times #math.equation(block: false, alt: "b")[$b$] #linebreak() the product of #math.equation(block: false, alt: "a")[$a$] and #math.equation(block: false, alt: "b")[$b$]], [#math.equation(block: false, alt: "a times b")[$a ⋅ b$], #math.equation(block: false, alt: "a b")[$a b$], #math.equation(block: false, alt: "a open parenthesis b close parenthesis")[$a ( b )$], #math.equation(block: false, alt: "open parenthesis a close parenthesis open parenthesis b close parenthesis")[$( a ) ( b )$]], [#strong[Division]], [#math.equation(block: false, alt: "a")[$a$] divided by #math.equation(block: false, alt: "b")[$b$] #linebreak() the quotient of #math.equation(block: false, alt: "a")[$a$] and #math.equation(block: false, alt: "b")[$b$] #linebreak() the ratio of #math.equation(block: false, alt: "a")[$a$] and #math.equation(block: false, alt: "b")[$b$] #linebreak() #math.equation(block: false, alt: "b")[$b$] divided into #math.equation(block: false, alt: "a")[$a$]], [#math.equation(block: false, alt: "a divided by b")[$a ÷ b$], #math.equation(block: false, alt: "a / b")[$a / b$], #math.equation(block: false, alt: "the fraction a over b")[$frac(a, b)$], #math.equation(block: false, alt: "b a")[$b ") " overline(a)$]], )) Look closely at these phrases using the four operations: - the sum #emph[of] #math.equation(block: false, alt: "a")[$a$] #emph[and] #math.equation(block: false, alt: "b")[$b$] - the difference #emph[of] #math.equation(block: false, alt: "a")[$a$] #emph[and] #math.equation(block: false, alt: "b")[$b$] - the product #emph[of] #math.equation(block: false, alt: "a")[$a$] #emph[and] #math.equation(block: false, alt: "b")[$b$] - the quotient #emph[of] #math.equation(block: false, alt: "a")[$a$] #emph[and] #math.equation(block: false, alt: "b")[$b$] Each phrase tells you to operate on two numbers. Look for the words #strong[#emph[of]] and #strong[#emph[and]] to find the numbers. #examplebox("Example 11")[][ Translate each word phrase into an algebraic expression: + ⓐ the difference of #math.equation(block: false, alt: "20")[$20$] and #math.equation(block: false, alt: "4")[$4$] + ⓑ the quotient of #math.equation(block: false, alt: "10 x")[$10 x$] and #math.equation(block: false, alt: "3")[$3$] #solutionbox[ ⓐ The key word is #emph[difference], which tells us the operation is subtraction. Look for the words #emph[of] and #emph[and] to find the numbers to subtract. #math.equation(block: true, alt: "the difference of 20 a n d 4; 20 minus 4; 20 minus 4")[$"the difference" #h(0.2em) "of" #h(0.2em) 20 #h(0.2em) a n d #h(0.2em) 4 \ 20 #h(0.2em) "minus" #h(0.2em) 4 \ 20 − 4$] ⓑ The key word is #emph[quotient], which tells us the operation is division. #math.equation(block: true, alt: "the quotient of 10 x and 3; divide 10 x by 3; 10 x divided by 3")[$"the quotient of" #h(0.2em) 10 x #h(0.2em) "and" #h(0.2em) 3 \ "divide" #h(0.2em) 10 x #h(0.2em) "by" #h(0.2em) 3 \ 10 x ÷ 3$] This can also be written as #math.equation(block: false, alt: "10 x / 3 or the fraction 10 x over 3")[$10 x / 3 #h(0.2em) "or" #h(0.4em) frac(10 x, 3)$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Translate the given word phrase into an algebraic expression: + ⓐ the difference of #math.equation(block: false, alt: "47")[$47$] and #math.equation(block: false, alt: "41")[$41$] + ⓑ the quotient of #math.equation(block: false, alt: "5 x")[$5 x$] and #math.equation(block: false, alt: "2")[$2$] #solutionbox[ + ⓐ 47 − 41 + ⓑ 5#emph[x] ÷ 2 ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Translate the given word phrase into an algebraic expression: + ⓐ the sum of #math.equation(block: false, alt: "17")[$17$] and #math.equation(block: false, alt: "19")[$19$] + ⓑ the product of #math.equation(block: false, alt: "7")[$7$] and #math.equation(block: false, alt: "x")[$x$] #solutionbox[ + ⓐ 17 + 19 + ⓑ 7#emph[x] ] ] How old will you be in eight years? What age is eight more years than your age now? Did you add #math.equation(block: false, alt: "8")[$8$] to your present age? Eight #emph[more than] means eight added to your present age. How old were you seven years ago? This is seven years less than your age now. You subtract #math.equation(block: false, alt: "7")[$7$] from your present age. Seven #emph[less than] means seven subtracted from your present age. #examplebox("Example 12")[][ Translate each word phrase into an algebraic expression: + ⓐ Eight more than #math.equation(block: false, alt: "y")[$y$] + ⓑ Seven less than #math.equation(block: false, alt: "9 z")[$9 z$] #solutionbox[ ⓐ The key words are #emph[more than]. They tell us the operation is addition. #emph[More than] means “added to”. #math.equation(block: true, alt: "Eight more than y; Eight added to y; y plus 8")[$"Eight more than" #h(0.2em) y \ "Eight added to" #h(0.2em) y \ y + 8$] ⓑ The key words are #emph[less than]. They tell us the operation is subtraction. #emph[Less than] means “subtracted from”. #math.equation(block: true, alt: "Seven less than 9 z; Seven subtracted from 9 z; 9 z minus 7")[$"Seven less than" #h(0.2em) 9 z \ "Seven subtracted from" #h(0.2em) 9 z \ 9 z − 7$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Translate each word phrase into an algebraic expression: + ⓐ Eleven more than #math.equation(block: false, alt: "x")[$x$] + ⓑ Fourteen less than #math.equation(block: false, alt: "11 a")[$11 a$] #solutionbox[ + ⓐ #emph[x] + 11 + ⓑ 11#emph[a] − 14 ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Translate each word phrase into an algebraic expression: + ⓐ #math.equation(block: false, alt: "19")[$19$] more than #math.equation(block: false, alt: "j")[$j$] + ⓑ #math.equation(block: false, alt: "21")[$21$] less than #math.equation(block: false, alt: "2 x")[$2 x$] #solutionbox[ + ⓐ #emph[j] + 19 + ⓑ 2#emph[x] − 21 ] ] #examplebox("Example 13")[][ Translate each word phrase into an algebraic expression: + ⓐ five times the sum of #math.equation(block: false, alt: "m")[$m$] and #math.equation(block: false, alt: "n")[$n$] + ⓑ the sum of five times #math.equation(block: false, alt: "m")[$m$] and #math.equation(block: false, alt: "n")[$n$] #solutionbox[ ⓐ There are two operation words: #emph[times] tells us to multiply and #emph[sum] tells us to add. Because we are multiplying #math.equation(block: false, alt: "5")[$5$] times the sum, we need parentheses around the sum of #math.equation(block: false, alt: "m")[$m$] and #math.equation(block: false, alt: "n .")[$n .$] five times the sum of #math.equation(block: false, alt: "m")[$m$] and #math.equation(block: false, alt: "n")[$n$] #linebreak() #math.equation(block: false, alt: "5 open parenthesis m plus n close parenthesis")[$#h(4em) 5 ( m + n )$] ⓑ To take a sum, we look for the words #emph[of] and #emph[and] to see what is being added. Here we are taking the sum #emph[of] five times #math.equation(block: false, alt: "m")[$m$] and #math.equation(block: false, alt: "n .")[$n .$] the sum of five times #math.equation(block: false, alt: "m")[$m$] and #math.equation(block: false, alt: "n")[$n$] #linebreak() #math.equation(block: false, alt: "5 m plus n")[$#h(4em) 5 m + n$] Notice how the use of parentheses changes the result. In part ⓐ , we add first and in part ⓑ , we multiply first. ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Translate the word phrase into an algebraic expression: + ⓐ four times the sum of #math.equation(block: false, alt: "p")[$p$] and #math.equation(block: false, alt: "q")[$q$] + ⓑ the sum of four times #math.equation(block: false, alt: "p")[$p$] and #math.equation(block: false, alt: "q")[$q$] #solutionbox[ + ⓐ 4(#emph[p] + #emph[q]) + ⓐ 4#emph[p] + #emph[q] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Translate the word phrase into an algebraic expression: + ⓐ the difference of two times #math.equation(block: false, alt: "x and 8")[$x #h(0.2em) "and 8" #h(0.2em)$] + ⓑ two times the difference of #math.equation(block: false, alt: "x and 8")[$x #h(0.2em) "and" #h(0.2em) 8$] #solutionbox[ + ⓐ 2#emph[x] − 8 + ⓑ 2(#emph[x] − 8) ] ] Later in this course, we’ll apply our skills in algebra to solving equations. We’ll usually start by translating a word phrase to an #strong[algebraic expression]. We’ll need to be clear about what the expression will represent. We’ll see how to do this in the next two examples. #examplebox("Example 14")[][ The height of a rectangular window is #math.equation(block: false, alt: "6")[$6$] inches less than the width. Let #math.equation(block: false, alt: "w")[$w$] represent the width of the window. Write an expression for the height of the window. #solutionbox[ #figure(table( columns: 2, align: left, inset: 6pt, table.header([Write a phrase about the height.], [#math.equation(block: false, alt: "6")[$6$] less than the width]), [Substitute #math.equation(block: false, alt: "w")[$w$] for the width.], [#math.equation(block: false, alt: "6")[$6$] less than #math.equation(block: false, alt: "w")[$w$]], [Rewrite 'less than' as 'subtracted from'.], [#math.equation(block: false, alt: "6")[$6$] subtracted from #math.equation(block: false, alt: "w")[$w$]], [Translate the phrase into algebra.], [#math.equation(block: false, alt: "w minus 6")[$w − 6$]], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ The length of a rectangle is #math.equation(block: false, alt: "5")[$5$] inches less than the width. Let #math.equation(block: false, alt: "w")[$w$] represent the width of the rectangle. Write an expression for the length of the rectangle. #solutionbox[ #emph[w] − 5 ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ The width of a rectangle is #math.equation(block: false, alt: "2")[$2$] meters greater than the length. Let #math.equation(block: false, alt: "l")[$l$] represent the length of the rectangle. Write an expression for the width of the rectangle. #solutionbox[ #emph[l] + 2 ] ] #examplebox("Example 15")[][ Blanca has dimes and quarters in her purse. The number of dimes is #math.equation(block: false, alt: "2")[$2$] less than #math.equation(block: false, alt: "5")[$5$] times the number of quarters. Let #math.equation(block: false, alt: "q")[$q$] represent the number of quarters. Write an expression for the number of dimes. #solutionbox[ #figure(table( columns: 2, align: left, inset: 6pt, table.header([Write a phrase about the number of dimes.], [two less than five times the number of quarters]), [Substitute #math.equation(block: false, alt: "q")[$q$] for the number of quarters.], [#math.equation(block: false, alt: "2")[$2$] less than five times #math.equation(block: false, alt: "q")[$q$]], [Translate #math.equation(block: false, alt: "5")[$5$] #emph[times] #math.equation(block: false, alt: "q")[$q$].], [#math.equation(block: false, alt: "2")[$2$] less than #math.equation(block: false, alt: "5 q")[$5 q$]], [Translate the phrase into algebra.], [#math.equation(block: false, alt: "5 q minus 2")[$5 q − 2$]], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Geoffrey has dimes and quarters in his pocket. The number of dimes is seven less than six times the number of quarters. Let #math.equation(block: false, alt: "q")[$q$] represent the number of quarters. Write an expression for the number of dimes. #solutionbox[ 6#emph[q] − 7 ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Lauren has dimes and nickels in her purse. The number of dimes is eight more than four times the number of nickels. Let #math.equation(block: false, alt: "n")[$n$] represent the number of nickels. Write an expression for the number of dimes. #solutionbox[ 4#emph[n] + 8 ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[ACCESS ADDITIONAL ONLINE RESOURCES] - #link("https://openstaxcollege.org/l/24AlgExpvocab")[Algebraic Expression Vocabulary] ] === Key Concepts - #emph[Combine like terms.] + Identify like terms. + Rearrange the expression so like terms are together. + Add the coefficients of the like terms ==== Practice Makes Perfect #strong[Evaluate Algebraic Expressions] In the following exercises, evaluate the expression for the given value. #math.equation(block: true, alt: "7 x plus 8 when x equals 2")[$7 x + 8 #h(0.2em) "when" #h(0.2em) x = 2$] #solutionbox[ 22 ] #math.equation(block: true, alt: "9 x plus 7 when x equals 3")[$9 x + 7 #h(0.2em) "when" #h(0.2em) x = 3$] #math.equation(block: true, alt: "5 x minus 4 when x equals 6")[$5 x − 4 #h(0.2em) "when" #h(0.2em) x = 6$] #solutionbox[ 26 ] #math.equation(block: true, alt: "8 x minus 6 when x equals 7")[$8 x − 6 #h(0.2em) "when" #h(0.2em) x = 7$] #math.equation(block: true, alt: "x squared when x equals 12")[$x^(2) #h(0.2em) "when" #h(0.2em) x = 12$] #solutionbox[ 144 ] #math.equation(block: true, alt: "x cubed when x equals 5")[$x^(3) #h(0.2em) "when" #h(0.2em) x = 5$] #math.equation(block: true, alt: "x to the power 5 when x equals 2")[$x^(5) #h(0.2em) "when" #h(0.2em) x = 2$] #solutionbox[ 32 ] #math.equation(block: true, alt: "x to the power 4 when x equals 3")[$x^(4) #h(0.2em) "when" #h(0.2em) x = 3$] #math.equation(block: true, alt: "3 to the power x when x equals 3")[$3^(x) #h(0.2em) "when" #h(0.2em) x = 3$] #solutionbox[ 27 ] #math.equation(block: true, alt: "4 to the power x when x equals 2")[$4^(x) #h(0.2em) "when" #h(0.2em) x = 2$] #math.equation(block: true, alt: "x squared plus 3 x minus 7 when x equals 4")[$x^(2) + 3 x − 7 #h(0.2em) "when" #h(0.2em) x = 4$] #solutionbox[ 21 ] #math.equation(block: true, alt: "x squared plus 5 x minus 8 when x equals 6")[$x^(2) + 5 x − 8 #h(0.2em) "when" #h(0.2em) x = 6$] #math.equation(block: true, alt: "2 x plus 4 y minus 5 when x equals 7 , y equals 8")[$2 x + 4 y − 5 #h(0.2em) "when" #h(0.2em) x = 7 , y = 8$] #solutionbox[ 41 ] #math.equation(block: true, alt: "6 x plus 3 y minus 9 when x equals 6 , y equals 9")[$6 x + 3 y − 9 #h(0.2em) "when" #h(0.2em) x = 6 , y = 9$] #math.equation(block: true, alt: "open parenthesis x minus y close parenthesis squared when x equals 10 , y equals 7")[$attach(( x − y ), t: 2) #h(0.2em) "when" #h(0.2em) x = 10 , y = 7$] #solutionbox[ 9 ] #math.equation(block: true, alt: "open parenthesis x plus y close parenthesis squared when x equals 6 , y equals 9")[$attach(( x + y ), t: 2) #h(0.2em) "when" #h(0.2em) x = 6 , y = 9$] #solutionbox[ 225 ] #math.equation(block: true, alt: "a squared plus b squared when a equals 3 , b equals 8")[$a^(2) + b^(2) #h(0.2em) "when" #h(0.2em) a = 3 , b = 8$] #solutionbox[ 73 ] #math.equation(block: true, alt: "r squared minus s squared when r equals 12 , s equals 5")[$r^(2) − s^(2) #h(0.2em) "when" #h(0.2em) r = 12 , s = 5$] #math.equation(block: true, alt: "2 l plus 2 w when l equals 15 , w equals 12")[$2 l + 2 w #h(0.2em) "when" #h(0.2em) l = 15 , w = 12$] #solutionbox[ 54 ] #math.equation(block: true, alt: "2 l plus 2 w when l equals 18 , w equals 14")[$2 l + 2 w #h(0.2em) "when" #h(0.2em) l = 18 , w = 14$] #strong[Identify Terms, Coefficients, and Like Terms] In the following exercises, list the terms in the given expression. #math.equation(block: true, alt: "15 x squared plus 6 x plus 2")[$15 x^(2) + 6 x + 2$] #solutionbox[ 15#emph[x]#super[2], 6#emph[x], 2 ] #math.equation(block: true, alt: "11 x squared plus 8 x plus 5")[$11 x^(2) + 8 x + 5$] #math.equation(block: true, alt: "10 y cubed plus y plus 2")[$10 y^(3) + y + 2$] #solutionbox[ 10#emph[y]#super[3], #emph[y], 2 ] #math.equation(block: true, alt: "9 y cubed plus y plus 5")[$9 y^(3) + y + 5$] In the following exercises, identify the coefficient of the given term. #math.equation(block: true, alt: "8 a")[$8 a$] #solutionbox[ 8 ] #math.equation(block: true, alt: "13 m")[$13 m$] #math.equation(block: true, alt: "5 r squared")[$5 r^(2)$] #solutionbox[ 5 ] #math.equation(block: true, alt: "6 x cubed")[$6 x^(3)$] In the following exercises, identify all sets of like terms. #math.equation(block: true, alt: "x cubed , 8 x , 14 , 8 y , 5 , 8 x cubed")[$x^(3) , 8 x , 14 , 8 y , 5 , 8 x^(3)$] #solutionbox[ #emph[x]#super[3] and 8#emph[x]#super[3]; 14 and 5 ] #math.equation(block: true, alt: "6 z , 3 w squared , 1 , 6 z squared , 4 z , w squared")[$6 z , 3 w^(2) , 1 , 6 z^(2) , 4 z , w^(2)$] #math.equation(block: true, alt: "9 a , a squared , 16 a b , 16 b squared , 4 a b , 9 b squared")[$9 a , a^(2) , 16 a b , 16 b^(2) , 4 a b , 9 b^(2)$] #solutionbox[ 16#emph[ab] and 4#emph[ab]; 16#emph[b]#super[2] and 9#emph[b]#super[2] ] #math.equation(block: true, alt: "3 , 25 r squared , 10 s , 10 r , 4 r squared , 3 s")[$3 , 25 r^(2) , 10 s , 10 r , 4 r^(2) , 3 s$] #strong[Simplify Expressions by Combining Like Terms] In the following exercises, simplify the given expression by combining like terms. #math.equation(block: true, alt: "10 x plus 3 x")[$10 x + 3 x$] #solutionbox[ 13#emph[x] ] #math.equation(block: true, alt: "15 x plus 4 x")[$15 x + 4 x$] #math.equation(block: true, alt: "17 a plus 9 a")[$17 a + 9 a$] #solutionbox[ 26#emph[a] ] #math.equation(block: true, alt: "18 z plus 9 z")[$18 z + 9 z$] #math.equation(block: true, alt: "4 c plus 2 c plus c")[$4 c + 2 c + c$] #solutionbox[ 7#emph[c] ] #math.equation(block: true, alt: "6 y plus 4 y plus y")[$6 y + 4 y + y$] #math.equation(block: true, alt: "9 x plus 3 x plus 8")[$9 x + 3 x + 8$] #solutionbox[ 12#emph[x] + 8 ] #math.equation(block: true, alt: "8 a plus 5 a plus 9")[$8 a + 5 a + 9$] #math.equation(block: true, alt: "7 u plus 2 plus 3 u plus 1")[$7 u + 2 + 3 u + 1$] #solutionbox[ 10#emph[u] + 3 ] #math.equation(block: true, alt: "8 d plus 6 plus 2 d plus 5")[$8 d + 6 + 2 d + 5$] #math.equation(block: true, alt: "7 p plus 6 plus 5 p plus 4")[$7 p + 6 + 5 p + 4$] #solutionbox[ 12#emph[p] + 10 ] #math.equation(block: true, alt: "8 x plus 7 plus 4 x minus 5")[$8 x + 7 + 4 x − 5$] #math.equation(block: true, alt: "10 a plus 7 plus 5 a minus 2 plus 7 a minus 4")[$10 a + 7 + 5 a − 2 + 7 a − 4$] #solutionbox[ 22#emph[a] + 1 ] #math.equation(block: true, alt: "7 c plus 4 plus 6 c minus 3 plus 9 c minus 1")[$7 c + 4 + 6 c − 3 + 9 c − 1$] #math.equation(block: true, alt: "3 x squared plus 12 x plus 11 plus 14 x squared plus 8 x plus 5")[$3 x^(2) + 12 x + 11 + 14 x^(2) + 8 x + 5$] #solutionbox[ 17#emph[x]#super[2] + 20#emph[x] + 16 ] #math.equation(block: true, alt: "5 b squared plus 9 b plus 10 plus 2 b squared plus 3 b minus 4")[$5 b^(2) + 9 b + 10 + 2 b^(2) + 3 b − 4$] #strong[Translate English Phrases into Algebraic Expressions] In the following exercises, translate the given word phrase into an algebraic expression. The sum of 8 and 12 #solutionbox[ 8 + 12 ] The sum of 9 and 1 The difference of 14 and 9 #solutionbox[ 14 − 9 ] 8 less than 19 The product of 9 and 7 #solutionbox[ 9 ⋅ 7 ] The product of 8 and 7 The quotient of 36 and 9 #solutionbox[ 36 ÷ 9 ] The quotient of 42 and 7 The difference of #math.equation(block: false, alt: "x")[$x$] and #math.equation(block: false, alt: "4")[$4$] #solutionbox[ #emph[x] − 4 ] #math.equation(block: false, alt: "3")[$3$] less than #math.equation(block: false, alt: "x")[$x$] The product of #math.equation(block: false, alt: "6")[$6$] and #math.equation(block: false, alt: "y")[$y$] #solutionbox[ 6#emph[y] ] The product of #math.equation(block: false, alt: "9")[$9$] and #math.equation(block: false, alt: "y")[$y$] The sum of #math.equation(block: false, alt: "8 x")[$8 x$] and #math.equation(block: false, alt: "3 x")[$3 x$] #solutionbox[ 8#emph[x] + 3#emph[x] ] The sum of #math.equation(block: false, alt: "13 x")[$13 x$] and #math.equation(block: false, alt: "3 x")[$3 x$] The quotient of #math.equation(block: false, alt: "y")[$y$] and #math.equation(block: false, alt: "3")[$3$] #solutionbox[ #emph[y] ÷ 3 ] The quotient of #math.equation(block: false, alt: "y")[$y$] and #math.equation(block: false, alt: "8")[$8$] Eight times the difference of #math.equation(block: false, alt: "y")[$y$] and nine #solutionbox[ 8 (#emph[y] − 9) ] Seven times the difference of #math.equation(block: false, alt: "y")[$y$] and one Five times the sum of #math.equation(block: false, alt: "x")[$x$] and #math.equation(block: false, alt: "y")[$y$] #solutionbox[ 5 (#emph[x] + #emph[y]) ] Nine times five less than twice #math.equation(block: false, alt: "x")[$x$] In the following exercises, write an algebraic expression. Adele bought a skirt and a blouse. The skirt cost #math.equation(block: false, alt: "$15")[$"$15"$] more than the blouse. Let #math.equation(block: false, alt: "b")[$b$] represent the cost of the blouse. Write an expression for the cost of the skirt. #solutionbox[ #emph[b] + 15 ] Eric has rock and classical CDs in his car. The number of rock CDs is #math.equation(block: false, alt: "3")[$3$] more than the number of classical CDs. Let #math.equation(block: false, alt: "c")[$c$] represent the number of classical CDs. Write an expression for the number of rock CDs. The number of girls in a second-grade class is #math.equation(block: false, alt: "4")[$4$] less than the number of boys. Let #math.equation(block: false, alt: "b")[$b$] represent the number of boys. Write an expression for the number of girls. #solutionbox[ #emph[b] − 4 ] Marcella has #math.equation(block: false, alt: "6")[$6$] fewer male cousins than female cousins. Let #math.equation(block: false, alt: "f")[$f$] represent the number of female cousins. Write an expression for the number of boy cousins. Greg has nickels and pennies in his pocket. The number of pennies is seven less than twice the number of nickels. Let #math.equation(block: false, alt: "n")[$n$] represent the number of nickels. Write an expression for the number of pennies. #solutionbox[ 2#emph[n] − 7 ] Jeannette has #math.equation(block: false, alt: "$5")[$"$5"$] and #math.equation(block: false, alt: "$10")[$"$10"$] bills in her wallet. The number of fives is three more than six times the number of tens. Let #math.equation(block: false, alt: "t")[$t$] represent the number of tens. Write an expression for the number of fives. ==== Everyday Math In the following exercises, use algebraic expressions to solve the problem. #strong[Car insurance] Justin’s car insurance has a #math.equation(block: false, alt: "$750")[$"$750"$] deductible per incident. This means that he pays #math.equation(block: false, alt: "$750")[$"$750"$] and his insurance company will pay all costs beyond #math.equation(block: false, alt: "$750.")[$"$750."$] If Justin files a claim for #math.equation(block: false, alt: "$2,100,")[$"$2,100,"$] how much will he pay, and how much will his insurance company pay? #solutionbox[ He will pay \$750. His insurance company will pay \$1350. ] #strong[Home insurance] Pam and Armando’s home insurance has a #math.equation(block: false, alt: "$2,500")[$"$2,500"$] deductible per incident. This means that they pay #math.equation(block: false, alt: "$2,500")[$"$2,500"$] and their insurance company will pay all costs beyond #math.equation(block: false, alt: "$2,500.")[$"$2,500."$] If Pam and Armando file a claim for #math.equation(block: false, alt: "$19,400,")[$"$19,400,"$] how much will they pay, and how much will their insurance company pay? ==== Writing Exercises Explain why “the sum of #emph[x] and #emph[y]” is the same as “the sum of #emph[y] and #emph[x],” but “the difference of #emph[x] and #emph[y]” is not the same as “the difference of #emph[y] and #emph[x].” Try substituting two random numbers for #math.equation(block: false, alt: "x")[$x$] and #math.equation(block: false, alt: "y")[$y$] to help you explain. Explain the difference between #math.equation(block: false, alt: "“4")[$"“4"$] times the sum of #math.equation(block: false, alt: "x")[$x$] and #math.equation(block: false, alt: "y ”")[$y "”"$] and “the sum of #math.equation(block: false, alt: "4")[$4$] times #math.equation(block: false, alt: "x")[$x$] and #math.equation(block: false, alt: "y .”")[$y ".”"$] ==== Self Check ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section. #figure(figph[A self-assessment table for algebraic skills. It includes columns for 'I can...', 'Confidently', 'With some help', and 'No-I don't get it!'. Skills listed are evaluating expressions, identifying terms, simplifying, and translating word phrases.], alt: "A self-assessment table for algebraic skills. It includes columns for 'I can...', 'Confidently', 'With some help', and 'No-I don't get it!'. Skills listed are evaluating expressions, identifying terms, simplifying, and translating word phrases.", caption: none) ⓑ After reviewing this checklist, what will you do to become confident for all objectives?