8.6 Chapter Summary and Review

Key Concepts

We can solve a $2 times 2$ linear system by graphing. The solution is the intersection point of the two graphs.
A linear system may be inconsistent (has no solution), dependent (has infinitely many solutions), or consistent and independent (has one solution).
Note
Inconsistent and Dependent Systems
1. If an equation of the form
  $0 x + 0 y = k (k \neq 0)$
  is obtained as a linear combination of the equations in a system, the system is inconsistent.
2. If an equation of the form
  $0 x + 0 y = 0$
  is obtained as a linear combination of the equations in a system, the system is dependent.
We can use a system of equations to solve problems involving two unknown quantities.
In economics, the price at which the supply and demand are equal is called the equilibrium price.
The solution to a $3 times 3$ linear system is an ordered triple.
A $3 times 3$ system in triangular form can be solved by back-substitution.
Gaussian reduction is a generalized form of the elimination method that can be used to reduce a $3 times 3$ linear system to triangular form.
Note
Steps for Solving a $3 times 3$ Linear System
1. Clear each equation of fractions and put it in standard form.
2. Choose two of the equations and eliminate one of the variables by forming a linear combination.
3. Choose a different pair of equations and eliminate the same variable.
4. Form a $2 times 2$ system with the equations found in steps (2) and (3). Eliminate one of the variables from this $2 times 2$ system by using a linear combination.
5. Form a triangular system by choosing among the previous equations. Use back-substitution to solve the triangular system.
$3 times 3$ linear systems may be inconsistent or dependent.
We can use a matrix to represent a system of linear equations. Each row of the matrix consists of the coefficients in one of the equations of the system.
We operate on a matrix by using the elementary row operations.
Note
Elementary Row Operations
1. Multiply the entries of any row by a nonzero real number.
2. Add a constant multiple of one row to another row.
3. Interchange two rows.
We can solve a linear system by matrix reduction.
Note
Solving a Linear System by Matrix Reduction
1. Write the augmented matrix for the system.
2. Using elementary row operations, transform the matrix into an equivalent one in upper triangular form.
3. Use back-substitution to find the solution to the system.
Note
Strategy for Matrix Reduction
1. If the first entry in the first row is zero, interchange that row with another. Obtain zeros in the first entries of the second and third rows by adding suitable multiples of the first row to the second and third rows.
2. If the second entry of the second row is zero, interchange the second and third rows. Obtain a zero in the second entry of the third row by adding a suitable multiple of the second row to the third row.
To reduce larger matrices, we start with the first row and work our way along the diagonal, using row operations to obtain nonzero entries on the diagonal and zeros below the diagonal entry.
The solutions of a linear inequality in two variables consist of a half-plane on one side of the line. The line itself is not included if the inequality is strict.
Once we have graphed the boundary line, we can decide which half-plane to shade by using a test point.
Note
To Graph an Inequality Using a Test Point:
1. Graph the corresponding equation to obtain the boundary line.
2. Choose a test point that does not lie on the boundary line.
3. Substitute the coordinates of the test point into the inequality.
  1. If the resulting statement is true, shade the half-plane that includes the test point.
  2. If the resulting statement is false, shade the half-plane that does not include the test point.
4. If the inequality is strict, make the boundary line a dashed line.
The solutions to a system of inequalities include all points that are solutions to each inequality in the system. The graph of the system is the intersection of the shaded regions for each inequality in the system
To describe the solutions of a system of inequalities, it is useful to locate the vertices, or corner points, of the boundary.
Linear programming is a technique for finding the maximum or minimum value of an objective function, subject to a system of constraints.
The optimum solution occurs at one of the vertices of the set of feasible solutions.
Note
To Solve a Linear Programming Problem:
1. Represent the unknown quantities by variables. Write the objective function and the constraints in terms of the variables.
2. Graph the solutions to the constraint inequalities.
3. Find the coordinates of each vertex of the solution set.
4. Evaluate the objective function at each vertex.
5. The maximum and minimum values of the objective function occur at vertices of the set of feasible solutions.

Chapter 8 Review Problems

For Problems 1–2, solve the system by graphing. Use the ZDecimal window.

$\begin{aligned} y & = - 2.9 x - 0.9 \\ y & = 1.4 - 0.6 x \end{aligned}$

$open parenthesis minus 1, 2 close parenthesis$

$\begin{aligned} y & = 0.6 x - 1.94 \\ y & = - 1.1 x + 1.29 \end{aligned}$

For Problems 3–6, solve the system using substitution or elimination.

$\begin{aligned} x + 5 y & = 18 \\ x - y & = - 3 \end{aligned}$

$(\frac{1}{2}, \frac{7}{2})$

$\begin{aligned} x + 5 y & = 11 \\ 2 x + 3 y & = 8 \end{aligned}$

$\begin{aligned} \frac{2}{3} x - 3 y & = 8 \\ x + \frac{3}{4} y & = 12 \end{aligned}$

$open parenthesis 12, 0 close parenthesis$

$\begin{aligned} 3 x & = 5 y - 6 \\ 3 y & = 10 - 11 x \end{aligned}$

For Problems 7–10, decide whether the system is inconsistent, dependent, or consistent and independent.

$\begin{aligned} 2 x - 3 y & = 4 \\ x + 2 y & = 7 \end{aligned}$

Consistent and independent

$\begin{aligned} 2 x - 3 y & = 4 \\ 6 x - 9 y & = 4 \end{aligned}$

$\begin{aligned} 2 x - 3 y & = 4 \\ 6 x - 9 y & = 12 \end{aligned}$

Dependent

$\begin{aligned} x - y & = 6 \\ x + y & = 6 \end{aligned}$

For Problems 11–16, solve the system using Gaussian reduction.

$\begin{aligned} x & + & 3 y & - & z & = 3 \\ 2 x & - & y & + & 3 z & = 1 \\ 3 x & + & 2 y & + & z & = 5 \end{aligned}$

$open parenthesis 2, 0, minus 1 close parenthesis$

$\begin{aligned} x & + & y & + & z & = 2 \\ 3 x & - & y & + & z & = 4 \\ 2 x & + & y & + & 2 z & = 3 \end{aligned}$

$\begin{aligned} x + z & = 5 \\ y - z & = - 8 \\ 2 x + z & = 7 \end{aligned}$

$open parenthesis 2, minus 5, 3 close parenthesis$

$\begin{aligned} x & + & 4 y & + & 4 z & = 0 \\ 3 x & + & 2 y & + & z & = - 4 \\ 2 x & - & 4 y & + & z & = - 11 \end{aligned}$

$\begin{aligned} \frac{1}{2} x & + & y & + & z & = 3 \\ x & - & 2 y & - & \frac{1}{3} z & = - 5 \\ \frac{1}{2} x & - & 3 y & - & \frac{2}{3} z & = - 6 \end{aligned}$

$open parenthesis minus 2, 1, 3 close parenthesis$

$\begin{aligned} \frac{3}{4} x & - & \frac{1}{2} y & + & 6 z & = 2 \\ \frac{1}{2} x & + & y & - & \frac{3}{4} z & = 0 \\ \frac{1}{4} x & + & \frac{1}{2} y & - & \frac{1}{2} z & = 0 \end{aligned}$

For Problems 17–22, use matrix reduction to solve the system.

$\begin{aligned} x & - & 2 y & = 5 \\ 2 x & + & y & = 5 \end{aligned}$

$open parenthesis 3, minus 1 close parenthesis$

$\begin{aligned} 4 x & - & 3 y & = 16 \\ 2 x & + & y & = 8 \end{aligned}$

$\begin{aligned} 2 x & - & y & = 7 \\ 3 x & + & 2 y & = 14 \end{aligned}$

$open parenthesis 4, 1 close parenthesis$

$\begin{aligned} 2 x & - & y & + & 3 z & = - 6 \\ x & + & 2 y & - & z & = 7 \\ 3 x & + & y & + & z & = 2 \end{aligned}$

$\begin{aligned} x & + & 2 y & - & z & = - 3 \\ 2 x & - & 3 y & + & 2 z & = 2 \\ x & - & y & + & 4 z & = 7 \end{aligned}$

$open parenthesis minus 1, 0, 2 close parenthesis$

$\begin{aligned} x & + & y & + & z & = 1 \\ 2 x & - & y & - & z & = 2 \\ 2 x & - & y & + & 3 z & = 2 \end{aligned}$

For Problems 23–26, solve the system by finding the reduced row echelon form of the augmented matrix.

$\begin{aligned} 2 a & + & 3 b & - & 4 c & - & 5 d & = 3 \\ 2 a & - & 3 b & + & 4 c & - & 7 d & = - 11 \\ 3 a & + & b & - & 8 c & + & d & = 9 \\ 4 a & - & 7 b & - & 5 c & + & 3 d & = - 4 \end{aligned}$

$open parenthesis 4, 3, 1, 2 close parenthesis$

$\begin{aligned} - a & - & 2 b & + & 5 c & + & 2 d & = 15 \\ - 2 a & + & 3 b & + & 2 c & + & d & = 15 \\ 2 a & - & 4 b & + & 6 c & + & 9 d & = 20 \\ 6 a & + & 8 b & + & 7 c & - & 2 d & = 0 \end{aligned}$

$\begin{aligned} 2 a & - & b & - & 3 c & + & d & + & 5 e & = 7 \\ 4 a & + & 6 b & - & 3 c & - & d & + & e & = - 6 \\ 5 a & + & 2 b & - & 9 c & - & 4 d & + & 7 e & = 3 \\ 6 a & - & 2 b & + & 7 c & + & 2 d & - & 8 e & = 11 \\ 7 a & + & 8 b & + & 2 c & + & 6 d & + & e & = 2 \end{aligned}$

$(2, - 1, 5, - 3, 4)$

$\begin{aligned} a & - & 4 b & + & 2 c & + & 3 d & - & e & = 7 \\ a & + & 2 b & - & 5 c & + & 2 d & - & 3 e & = - 6 \\ 3 a & + & 3 b & + & 2 c & - & 4 d & + & 3 e & = 3 \\ - 2 a & - & 3 b & + & 5 c & + & 4 d & - & e & = 11 \\ - 4 a & + & 3 b & - & c & + & d & + & 2 e & = 2 \end{aligned}$

For Problems 27–32, solve the problem by writing and solving a system of linear equations in two or three variables.

A math contest exam has 40 questions. A contestant scores $5$ points for each correct answer but loses $2$ points for each wrong answer. Lupe answered all the questions and her score was $102$ . How many questions did she answer correctly?

$26$

A game show contestant wins $ $25$ for each correct answer he gives but loses $ $10$ for each incorrect response. Roger answered $24$ questions and won $ $355$ . How many answers did he get right?

Barbara wants to earn $ $500$ a year by investing $ $5000$ in two accounts, a savings plan that pays $8 %$ annual interest and a high-risk option that pays $13.5 %$ interest. How much should she invest in each account?

$ $3181.82$ at $8 %$ , $ $1818.18$ at $13.5 %$

An investment broker promises his client a $12 %$ return on her funds. If the broker invests $ $3000$ in bonds paying $8 %$ interest, how much must he invest in stocks paying $15 %$ interest to keep his promise?

The perimeter of a triangle is $30$ centimeters. The length of one side is $7$ centimeters shorter than the second side, and the third side is $1$ centimeter longer than the second side. Find the length of each side.

$5$ cm, $12$ cm, $13$ cm

A company ships its product to three cities: Boston, Chicago, and Los Angeles. The cost of shipping is $ $10$ per crate to Boston, $ $5$ per crate to Chicago, and $ $12$ per crate to Los Angeles. The company's shipping budget for April is $ $445$ . It has $55$ crates to ship, and demand for its product is twice as high in Boston as in Los Angeles. How many crates should the company ship to each destination?

For Problems 33–36, graph the inequality.

$3 x minus 4 y less than 12$

$x greater than 3 y minus 6$

$y < - \frac{1}{2}$

$minus 4 less than or equal to x less than 2$

For Problems 37–40, graph the solutions to the system of inequalities.

$y greater than 3, x less than or equal to 2$

$y greater than or equal to x, x greater than 2$

$3 x - y < 6, x + 2 y > 6$

$x - 3 y > 3, y < x + 2$

For Problems 41–44,

Graph the solutions to the system of inequalities.
Find the coordinates of the vertices.

$3 x minus 4 y less than or equal to 12; x greater than or equal to 0, y less than or equal to 0$

$2 x minus y less than or equal to 6; y less than or equal to x; x greater than or equal to 0, y greater than or equal to 0$

$x plus y less than or equal to 5; y greater than or equal to x; y greater than or equal to 2, x greater than or equal to 0$

$\begin{array}{r} x - y \leq - 3 \\ x + y \leq 6 \\ x \leq 4 \\ x \geq 0, y \geq 0 \end{array}$

Ruth wants to provide cookies for the customers at her bookstore. It takes $20$ minutes to mix the ingredients for each batch of peanut butter cookies and $10$ minutes to bake them. Each batch of granola cookies takes $8$ minutes to mix and $10$ minutes to bake. Ruth does not want to use the oven more than $2$ hours a day or to spend more than $2$ hours a day mixing ingredients. Write a system of inequalities for the number of batches of peanut butter cookies and of granola cookies that Ruth can make in one day and graph the solutions.

$20 p + 8 g \leq 120, 10 p + 10 g \leq 120$

A vegetarian recipe calls for no more than $32$ ounces of a combination of tofu and tempeh. Tofu provides $2$ grams of protein per ounce and tempeh provides $1.6$ grams of protein per ounce. Graham would like the dish to provide at least $56$ grams of protein. Write a system of inequalities for the amount of tofu and the amount of tempeh for the recipe and graph the solutions.

For Problems 47–48,

Graph the set of feasible solutions.
Find the vertex that gives the minimum of the objective function, and find the minimum value.
Find the vertex that gives the maximum of the objective function, and find the maximum value.

Objective function $C = 18 x + 48 y$ with constraints

$\begin{aligned} 3 x + y & \geq 3 \\ 2 x + y & \leq 12 \\ x + 5 y & \leq 15 \\ x \geq 0, y & \geq 0 \end{aligned}$

$P open parenthesis 1, 0 close parenthesis$ ; $18$
$Q open parenthesis 5, 2 close parenthesis$ ; $186$

Objective function $C = 10 x - 8 y$ with constraints

$\begin{aligned} 5 x - y & \geq 2 \\ x + 2 y & \leq 18 \\ x - y & \leq 3 \\ x \geq 0, y & \geq 0 \end{aligned}$

Write a system of inequalities for the number of batches of peanut butter cookies and granola cookies Ruth can make in one day and graph the solutions.
Ruth decides to sell the cookies. If she charges $25$ ¢ per peanut butter cookie and $20$ ¢ per granola cookie, she will sell all the cookies she bakes. Each batch contains $50$ cookies. How many batches of each type of cookie should she bake to maximize her income? What is the maximum income?

$20 p plus 8 g less than or equal to 120$ , $10 p plus 10 g less than or equal to 120$ , $p greater than or equal to 0$ , $g greater than or equal to 0$
$2$ batches peanut butter cookies; $10$ batches of granola cookies; for $ $125$

A vegetarian recipe calls for $32$ ounces of a combination of tofu and tempeh. Tofu provides $2$ grams of protein per ounce and tempeh provides $1.6$ grams of protein per ounce. Graham would like the dish to provide at least $56$ grams of protein.

Write a system of inequalities for the amount of tofu and the amount of tempeh for the recipe and graph the solutions.
Suppose that tofu costs $12$ ¢ per ounce and tempeh costs $16$ ¢ per ounce. What is the least expensive combination of tofu and tempeh Graham can use for the recipe? How much will it cost?

Modeling, Functions, and Graphs by Katherine Yoshiwara (yoshiwarabooks.org), GNU Free Documentation License 1.2 or later. Adapted for the XYZ HTML edition with the authors' permission (recorded 2026-07-04). License: GFDL-1.2-or-later.