Jay Abramson; Sharon North

Practice Test

Is the following ordered pair a solution to the system of equations?

1.

$\begin{array}{l} \begin{array}{l} - 5 x - y = 12 \\ x + 4 y = 9 \end{array} \end{array}$ with $(- 3, 3)$

For the following exercises, solve the systems of linear and nonlinear equations using substitution or elimination. Indicate if no solution exists.

2.

$\begin{array}{r} \frac{1}{2} x - \frac{1}{3} y = 4 \\ \frac{3}{2} x - y = 0 \end{array}$

3.

$\begin{array}{l} - \frac{1}{2} x - 4 y = 4 \\ 2 x + 16 y = 2 \end{array}$

4.

$\begin{array}{r} 5 x - y = 1 \\ - 10 x + 2 y = - 2 \end{array}$

5.

$\begin{array}{l} 4 x - 6 y - 2 z = \frac{1}{10} \\ x - 7 y + 5 z = - \frac{1}{4} \\ 3 x + 6 y - 9 z = \frac{6}{5} \end{array}$

6.

$\begin{array}{r} x + z = 20 \\ x + y + z = 20 \\ x + 2 y + z = 10 \end{array}$

7.

$\begin{array}{r} 5 x - 4 y - 3 z = 0 \\ 2 x + y + 2 z = 0 \\ x - 6 y - 7 z = 0 \end{array}$

8.

$\begin{array}{l} y = x^{2} + 2 x - 3 \\ y = x - 1 \end{array}$

9.

$\begin{array}{l} y^{2} + x^{2} = 25 \\ y^{2} - 2 x^{2} = 1 \end{array}$

For the following exercises, graph the following inequalities.

10.

$y < x^{2} + 9$

11.

$\begin{array}{l} x^{2} + y^{2} > 4 \\ y < x^{2} + 1 \end{array}$

For the following exercises, write the partial fraction decomposition.

12.

$\frac{- 8 x - 30}{x^{2} + 10 x + 25}$

13.

$\frac{13 x + 2}{{(3 x + 1)}^{2}}$

14.

$\frac{x^{4} - x^{3} + 2 x - 1}{x {(x^{2} + 1)}^{2}}$

For the following exercises, perform the given matrix operations.

15.

$5 [\begin{matrix} 4 & 9 \\ - 2 & 3 \end{matrix}] + \frac{1}{2} [\begin{matrix} - 6 & 12 \\ 4 & - 8 \end{matrix}]$

16.

$[\begin{array}{r} 1 & 4 & - 7 \\ - 2 & 9 & 5 \\ 12 & 0 & - 4 \end{array}] [\begin{matrix} 3 & - 4 \\ 1 & 3 \\ 5 & 10 \end{matrix}]$

17.

${[\begin{array}{r} \frac{1}{2} & \frac{1}{3} \\ \frac{1}{4} & \frac{1}{5} \end{array}]}^{- 1}$

18.

$\det | \begin{matrix} 0 & 0 \\ 400 & 4, 000 \end{matrix} |$

19.

$\det | \begin{array}{r} \frac{1}{2} & - \frac{1}{2} & 0 \\ - \frac{1}{2} & 0 & \frac{1}{2} \\ 0 & \frac{1}{2} & 0 \end{array} |$

20.

If $\det (A) = −6,$ what would be the determinant if you switched rows 1 and 3, multiplied the second row by 12, and took the inverse?

21.

Rewrite the system of linear equations as an augmented matrix.

\begin{array}{l} 14 x - 2 y + 13 z = 140 \\ - 2 x + 3 y - 6 z = - 1 \\ x - 5 y + 12 z = 11 \end{array}

22.

Rewrite the augmented matrix as a system of linear equations.

[\begin{array}{r} 1 & 0 & 3 \\ - 2 & 4 & 9 \\ - 6 & 1 & 2 \end{array} | \begin{array}{r} 12 \\ - 5 \\ 8 \end{array}]

For the following exercises, use Gaussian elimination to solve the systems of equations.

23.

$\begin{array}{r} x - 6 y = 4 \\ 2 x - 12 y = 0 \end{array}$

24.

$\begin{array}{r} 2 x + y + z = - 3 \\ x - 2 y + 3 z = 6 \\ x - y - z = 6 \end{array}$

For the following exercises, use the inverse of a matrix to solve the systems of equations.

25.

$\begin{array}{r} 4 x - 5 y = - 50 \\ - x + 2 y = 80 \end{array}$

26.

$\begin{array}{r} \frac{1}{100} x - \frac{3}{100} y + \frac{1}{20} z = - 49 \\ \frac{3}{100} x - \frac{7}{100} y - \frac{1}{100} z = 13 \\ \frac{9}{100} x - \frac{9}{100} y - \frac{9}{100} z = 99 \end{array}$

For the following exercises, use Cramer’s Rule to solve the systems of equations.

27.

$\begin{array}{l} 200 x - 300 y = 2 \\ 400 x + 715 y = 4 \end{array}$

28.

$\begin{array}{l} 0.1 x + 0.1 y - 0.1 z = - 1.2 \\ 0.1 x - 0.2 y + 0.4 z = - 1.2 \\ 0.5 x - 0.3 y + 0.8 z = - 5.9 \end{array}$

For the following exercises, solve using a system of linear equations.

29.

A factory producing cell phones has the following cost and revenue functions: $C (x) = x^{2} + 75 x + 2, 688$ and $R (x) = x^{2} + 160 x .$ What is the range of cell phones they should produce each day so there is profit? Round to the nearest number that generates profit.

30.

A small fair charges $1.50 for students, $1 for children, and $2 for adults. In one day, three times as many children as adults attended. A total of 800 tickets were sold for a total revenue of $1,050. How many of each type of ticket was sold?