Review Exercises and Sample Exam

3.8 Review Exercises and Sample Exam

Review Exercises

Linear Systems and Their Solutions

Determine whether or not the given ordered pair is a solution to the given system.

$(\frac{2}{3}, - 4)$ ;
${\begin{cases} 9 x - y = 10 \\ 3 x + 4 y = - 14 \end{cases}$
$(- \frac{1}{2}, \frac{3}{4})$ ;
${\begin{cases} 6 x - 8 y = - 9 \\ x + 2 y = 1 \end{cases}$
$(- 5, - \frac{7}{8})$ ;
${\begin{cases} x - 16 y = 9 \\ 2 x - 8 y = - 17 \end{cases}$
$(- 1, \frac{4}{5})$ ;
${\begin{cases} 2 x + 5 y = 2 \\ 3 x - 10 y = - 5 \end{cases}$

Given the graphs, determine the simultaneous solution.

Solve by graphing.

${\begin{cases} 2 x + y = 6 \\ x - 2 y = 8 \end{cases}$
${\begin{cases} 5 x - 2 y = 0 \\ x - y = 3 \end{cases}$
${\begin{cases} 4 x + 3 y = - 12 \\ - 8 x - 6 y = 24 \end{cases}$
${\begin{cases} \frac{1}{2} x + 2 y = 6 \\ x + 4 y = - 1 \end{cases}$
${\begin{cases} 5 x + 2 y = 30 \\ y - 5 = 0 \end{cases}$
${\begin{cases} 5 x + 3 y = - 15 \\ x + 3 = 0 \end{cases}$
${\begin{cases} \frac{1}{3} x - \frac{1}{2} y = 2 \\ \frac{1}{2} x + \frac{3}{5} y = 3 \end{cases}$
${\begin{cases} \frac{2}{5} x + \frac{1}{2} y = 1 \\ \frac{1}{15} x + \frac{1}{6} y = - \frac{1}{3} \end{cases}$

Solving Linear Systems with Two Variables

Solve by substitution.

${\begin{cases} 4 x - y = 12 \\ x + 3 y = - 10 \end{cases}$
${\begin{cases} 9 x - 2 y = 3 \\ x - 3 y = 17 \end{cases}$
${\begin{cases} 12 x + y = 7 \\ 3 x - 4 y = 6 \end{cases}$
${\begin{cases} 3 x - 2 y = 1 \\ 2 x + 3 y = - 1 \end{cases}$

Solve by elimination.

${\begin{cases} 5 x - 2 y = - 12 \\ 4 x + 6 y = - 21 \end{cases}$
${\begin{cases} 4 x - 5 y = 12 \\ 8 x + 3 y = - 2 \end{cases}$
${\begin{cases} 5 x - 3 y = 11 \\ 2 x - 4 y = - 4 \end{cases}$
${\begin{cases} 7 x + 2 y = 3 \\ 3 x + 5 y = - 7 \end{cases}$

Solve using any method.

${\begin{cases} 4 x - 8 y = 4 \\ x + 2 y = 9 \end{cases}$
${\begin{cases} 6 x - 9 y = 8 \\ x - y = 1 \end{cases}$
${\begin{cases} 2 x - 6 y = - 1 \\ 6 x + 10 y = - 3 \end{cases}$
${\begin{cases} 2 x - 3 y = 36 \\ x - 3 y = 9 \end{cases}$
${\begin{cases} 5 x - 3 y = 10 \\ - 10 x + 6 y = 3 \end{cases}$
${\begin{cases} \frac{1}{2} x - y = 3 \\ 3 x - 6 y = 18 \end{cases}$
${\begin{cases} \frac{3}{5} x - \frac{1}{2} y = - 1 \\ \frac{1}{10} x + \frac{3}{4} y = - 1 \end{cases}$
${\begin{cases} \frac{4}{3} x - \frac{2}{5} y = - \frac{8}{15} \\ \frac{1}{2} x - \frac{2}{3} y = - \frac{11}{24} \end{cases}$

Applications of Linear Systems with Two Variables

Set up a linear system and solve.

The sum of two integers is 32. The larger is 4 less than twice the smaller. Find the integers.
The sum of 2 times a larger integer and 3 times a smaller integer is 54. When twice the smaller integer is subtracted from the larger, the result is −1. Find the integers.
The length of a rectangle is 2 centimeters less than three times its width and the perimeter measures 44 centimeters. Find the dimensions of the rectangle.
The width of a rectangle is one-third of its length. If the perimeter measures $53 \frac{1}{3}$ centimeters, then find the dimensions of the rectangle.
The sum of a larger integer and 3 times a smaller is 61. When twice the smaller integer is subtracted from the larger, the result is 1. Find the integers.
A total of $8,600 was invested in two accounts. One account earned $4 \frac{3}{4}$ % annual interest and the other earned $6 \frac{1}{2}$ % annual interest. If the total interest for one year was $431.25, how much was invested in each account?
A jar consisting of only nickels and dimes contains 76 coins. If the total value is $6, how many of each coin are in the jar?
A nurse wishes to obtain 32 ounces of a 1.2% saline solution. How much of a 1% saline solution must she mix with a 2.6% saline solution to achieve the desired mixture?
A light aircraft flying with the wind can travel 330 miles in 2 hours. The aircraft can fly the same distance against the wind in 3 hours. Find the speed of the wind.
An executive was able to average 52 miles per hour to the airport in her car and then board an airplane that averaged 340 miles per hour. If the total 640-mile business trip took 4 hours, how long did she spend on the airplane?

Solving Linear Systems with Three Variables

Determine whether the given ordered triple is a solution to the given system.

$(- 2, - 1, 3)$ ;
${\begin{matrix} 4 x - y + 2 z = - 1 \\ x - 4 y + 3 z = 11 \\ 3 x + 5 y - 4 z = 1 \end{matrix}$
$(5, - 3, - 2)$ ;
${\begin{matrix} x - 4 y + 6 z = 5 \\ 2 x + 5 y - z = - 3 \\ 3 x - 4 y + z = 25 \end{matrix}$
$(1, - \frac{3}{2}, - \frac{4}{3})$ ;
${\begin{matrix} 5 x - 4 y + 3 z = 7 \\ x + 2 y - 6 z = 6 \\ 12 x - 6 y + 6 z = 13 \end{matrix}$
$(\frac{5}{4}, - \frac{1}{3}, 2)$ ;
${\begin{matrix} 8 x + 9 y + z = 9 \\ 4 x + 12 y - 4 z = - 7 \\ 12 x - 6 y - z = - 5 \end{matrix}$

Solve.

${\begin{matrix} 2 x + 3 y - z = 1 \\ 5 y + 2 z = 12 \\ 3 z = 18 \end{matrix}$
${\begin{matrix} 3 x - 5 y - 2 z = 21 \\ y - 7 z = 18 \\ 4 z = - 12 \end{matrix}$
${\begin{matrix} 4 x - 5 y - z = - 6 \\ 3 x + 6 y + 5 z = 3 \\ 5 x - 2 y - 3 z = - 17 \end{matrix}$
${\begin{matrix} x - 6 y + 3 z = - 2 \\ 5 x + 4 y - 2 z = 24 \\ 6 x - 8 y - 5 z = 25 \end{matrix}$
${\begin{matrix} x + 2 y - 2 z = 1 \\ 2 x - y - z = - 2 \\ 6 x - 3 y - 3 z = 12 \end{matrix}$
${\begin{matrix} 3 x + y + 2 z = - 1 \\ 9 x + 3 y + 6 z = - 3 \\ 4 x + y + 4 z = - 3 \end{matrix}$
${\begin{matrix} 3 a - 2 b + 5 c = - 3 \\ 6 a + 4 b - c = - 2 \\ - 6 a + 6 b + 24 c = 7 \end{matrix}$
${\begin{matrix} 9 a - 2 b - 6 c = 10 \\ 5 a - 3 b - 10 c = 14 \\ - 3 a + 4 b + 12 c = - 20 \end{matrix}$

Set up a linear system and solve.

The sum of three integers is 24. The larger is equal to the sum of the two smaller integers. Three times the smaller is equal to the larger. Find the integers.
The sports center sold 120 tickets to the Friday night basketball game for a total of $942. A general admission ticket cost $12, a student ticket cost $6, and a child ticket cost $4. If the sum of the general admission and student tickets totaled 105, then how many of each ticket were sold?
A 16-ounce mixed nut product containing 13.5% peanuts is to be packaged. The packager has a three-mixed nut product containing 6%, 10%, and 50% peanut concentrations in stock. If the amount of 50% peanut product is to be one-quarter that of the 10% peanut product, then how much of each will be needed to produce the desired peanut concentration?
Water is to be mixed with two acid solutions to produce a 25-ounce solution containing 6% acid. The acid mixtures on hand contain 10% and 25% acid. If the amount of 25% acid is to be one-half the amount of the 10% acid solution, how much water will be needed?

Matrices and Gaussian Elimination

Construct the corresponding augmented matrix.

${\begin{matrix} 9 x - 7 y = 4 \\ 3 x - y = - 1 \end{matrix}$
${\begin{matrix} x - 5 y = 12 \\ 3 y = - 5 \end{matrix}$
${\begin{matrix} x - y + 2 z = - 6 \\ 3 x - 6 y - z = 3 \\ - x + y - 5 z = 10 \end{matrix}$
${\begin{matrix} 5 x + 7 y - z = 0 \\ - 8 y + z = - 1 \\ - x + 3 z = - 9 \end{matrix}$

Solve using matrices and Gaussian elimination.

${\begin{matrix} 4 x + 5 y = 0 \\ 2 x - 3 y = 22 \end{matrix}$
${\begin{matrix} 3 x - 8 y = 20 \\ 2 x + 5 y = 3 \end{matrix}$
${\begin{matrix} x - y + 4 z = 1 \\ - 2 x + 3 y - 2 z = 0 \\ x - 6 y + 8 z = 8 \end{matrix}$
${\begin{matrix} - x + 3 y - z = 1 \\ 3 x - 6 y + 2 z = - 4 \\ 4 x - 3 y + 2 z = - 7 \end{matrix}$
${\begin{matrix} 5 x - 3 y - z = 2 \\ x - 6 y + z = 7 \\ 2 x + 6 y - 2 z = - 8 \end{matrix}$
${\begin{matrix} x + 2 y + 3 z = 4 \\ x + 3 y + z = 3 \\ 2 x + 5 y + 4 z = 8 \end{matrix}$
${\begin{matrix} 2 a + 5 b - c = 4 \\ 2 a + c = - 2 \\ a + b + 3 c = 6 \end{matrix}$
${\begin{matrix} a + 2 b + 3 c = - 7 \\ 4 b - 2 c = 8 \\ 3 a - c = - 7 \end{matrix}$

Determinants and Cramer’s Rule

Calculate the determinant.

$| \begin{matrix} - 9 & 5 \\ - 1 & 3 \end{matrix} |$
$| \begin{matrix} - 5 & 5 \\ - 3 & 3 \end{matrix} |$
$| \begin{matrix} 0 & 7 \\ 2 & 3 \end{matrix} |$
$| \begin{matrix} 0 & b_{1} \\ a_{2} & b_{2} \end{matrix} |$
$| \begin{array}{r} 2 & - 3 & 0 \\ 1 & - 2 & - 1 \\ 0 & 1 & 3 \end{array} |$
$| \begin{array}{r} 3 & 2 & - 1 \\ 1 & - 1 & 0 \\ 5 & - 2 & - 4 \end{array} |$
$| \begin{array}{r} 5 & - 3 & - 1 \\ 1 & - 6 & 1 \\ 2 & 6 & - 2 \end{array} |$
$| \begin{array}{r} a_{1} & 0 & 0 \\ a_{2} & b_{2} & 0 \\ a_{3} & b_{3} & c_{3} \end{array} |$

Solve using Cramer’s rule.

${\begin{cases} 2 x - 3 y = - 4 \\ 3 x + 5 y = 1 \end{cases}$
${\begin{matrix} 3 x - y = 2 \\ - 2 x + 6 y = 1 \end{matrix}$
${\begin{cases} 3 x + 5 y = 6 \\ 6 x + y = - 6 \end{cases}$
${\begin{cases} 6 x - 4 y = - 1 \\ - 3 x + 2 y = 2 \end{cases}$
${\begin{matrix} 5 x + 2 y + 4 z = 4 \\ 4 x + 3 y + 2 z = - 5 \\ - 5 x - 3 y - 5 z = 0 \end{matrix}$
${\begin{matrix} 2 x - y + 2 z = 1 \\ x - 3 y + z = 2 \\ 3 x - y - 4 z = - 2 \end{matrix}$
${\begin{matrix} 4 x - y - 2 z = - 7 \\ 2 x + y + 6 z = 0 \\ 2 x + 2 y + 4 z = - 1 \end{matrix}$
${\begin{matrix} x - y - z = 1 \\ 2 x - y + 3 z = 2 \\ x + y + z = - 1 \end{matrix}$
${\begin{matrix} 4 x - y + 2 z = - 1 \\ 2 x + 3 y - z = 3 \\ 6 x + 2 y + z = 2 \end{matrix}$
${\begin{matrix} x - y + 2 z = 1 \\ 2 x + 2 y - z = 2 \\ 3 x + y + z = 1 \end{matrix}$

Systems of Inequalities with Two Variables

Determine whether or not the given point is a solution to the system of inequalities.

(−6, 1);
${\begin{cases} - x + y > 2 \\ x - 2 y \leq - 1 \end{cases}$
$(\frac{1}{2}, - 3)$ ;
${\begin{cases} 4 x - 2 y \geq 8 \\ 6 x + 2 y < - 3 \end{cases}$
(−4, −2);
${\begin{cases} x - y > - 3 \\ 2 x + 3 y \leq 0 \\ - 3 x + 4 y \geq 4 \end{cases}$
$(5, - \frac{1}{5})$ ;
${\begin{cases} y < x^{2} - 25 \\ y > \frac{2}{3} x - 1 \end{cases}$
(−3, −2);
${\begin{cases} y < {(x - 1)}^{2} \\ y \leq | x + 1 | - 3 \end{cases}$
$(2, - \frac{2}{3})$ ;
${\begin{cases} y < 0 \\ x^{2} + y \geq 3 \end{cases}$

Graph the solution set.

${\begin{cases} y \leq - 4 \\ x - 2 y > 8 \end{cases}$
${\begin{cases} x + 4 y > 8 \\ 2 x - y \leq 4 \end{cases}$
${\begin{cases} y - 3 < 0 \\ - 2 x + 3 y > - 9 \\ x + y \geq 1 \end{cases}$
${\begin{cases} y \leq 0 \\ 2 x - 6 y < 9 \\ - 2 x + 6 y < 9 \end{cases}$
${\begin{cases} 2 x + y < 3 \\ y > {(x - 2)}^{2} - 5 \end{cases}$
${\begin{cases} y > | x | \\ y \geq - x^{2} + 6 \end{cases}$
${\begin{cases} x - 2 y < 12 \\ y \leq {(x - 4)}^{3} \end{cases}$
${\begin{cases} y + 6 > 0 \\ y < \sqrt{x} \end{cases}$

Answers

Yes
No
(−6, 2)
Ø
(4, −2)
$(x, - \frac{4}{3} x - 4)$
(4, 5)
(6, 0)

(2, −4)
$(\frac{2}{3}, - 1)$
$(- 3, - \frac{3}{2})$
(4, 3)
(5, 2)
$(- \frac{1}{2}, 0)$
Ø
$(- \frac{5}{2}, - 1)$

12, 20
Length: 16 centimeters; width: 6 centimeters
12, 25
The jar contains 32 nickels and 44 dimes.
27.5 miles per hour

No
Yes
$(\frac{7}{2}, 0, 6)$
(−2, −1, 3)
Ø
$(- \frac{2}{3}, \frac{1}{2}, 0)$
4, 8, 12
6 oz of the 6% peanut stock, 8 oz of the 10% peanut stock, and 2 oz of the 50% peanut stock should be mixed.

$[\begin{matrix} 9 & - 7 & | & 4 \\ 3 & - 1 & | & - 1 \end{matrix}]$
$[\begin{array}{r} 1 & - 1 & 2 | & - 6 \\ 3 & - 6 & - 1 | & 3 \\ - 1 & 1 & - 5 | & 10 \end{array}]$
(5, −4)
$(- 2, - 1, \frac{1}{2})$
$(x, \frac{2}{3} x - 1, 3 x + 1)$
(−2, 2, 2)

−22
−14
−1
0
$(- \frac{17}{19}, \frac{14}{19})$
$(- \frac{4}{3}, 2)$
(2, −5, 1)
$(- \frac{3}{2}, 0, \frac{1}{2})$
$(x, - \frac{8}{5} x + 1, - \frac{14}{5} x)$

Sample Exam

Determine whether or not $(- 2, \frac{3}{4})$ is a solution to ${\begin{cases} 2 x - 8 y = - 10 \\ 3 x + 4 y = - 3 \end{cases}$ .
Determine whether or not $(- 3, 2, - 5)$ is a solution to ${\begin{cases} x - y + 2 z = - 15 \\ 2 x - 3 y + z = - 17 \\ 3 x + 5 y - 2 z = 10 \end{cases}$ .