# 9.R: Systems of Equations and Inequalities (Review)

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## 9.1: Systems of Linear Equations: Two Variables

For the exercises 1-2, determine whether the ordered pair is a solution to the system of equations.

1) \begin{align*} 3x-y &= 4\\ x+4y &= -3 \end{align*}\; \; \text{ and }\; (-1,1)

No

2) \begin{align*} 6x-2y &= 24\\ -3x+3y &= 18 \end{align*}\; \; \text{ and }\; (9,15)

For the exercises 3-5, use substitution to solve the system of equations.

3) \begin{align*} 10x+5y &= -5\\ 3x-2y &= -12 \end{align*}

$$(-2,3)$$

4) \begin{align*} \dfrac{4}{7}x+\dfrac{1}{5}y &= \dfrac{43}{70}\\ \dfrac{5}{6}x-\dfrac{1}{3}y &= -\dfrac{2}{3} \end{align*}

5) \begin{align*} 5x+6y &= 14\\ 4x+8y &= 8 \end{align*}

$$(4,-1)$$

For the exercises 6-8, use addition to solve the system of equations.

6) \begin{align*} 3x+2y &= -7\\ 2x+4y &= 6 \end{align*}

7) \begin{align*} 3x+4y &= 2\\ 9x+12y &= 3 \end{align*}

No solutions exist.

8) \begin{align*} 8x+4y &= 2\\ 6x-5y &= 0.7 \end{align*}

For the exercises 9-10, write a system of equations to solve each problem. Solve the system of equations.

9) A factory has a cost of production $$C(x)=150x+15,000$$ and a revenue function $$R(x)=200x$$. What is the break-even point?

$$(300,60,000)$$

10) A performer charges $$C(x)=50x+10,000$$, where $$x$$ is the total number of attendees at a show. The venue charges $$\75$$ per ticket. After how many people buy tickets does the venue break even, and what is the value of the total tickets sold at that point?

$$(400,30,000)$$

## 9.2: Systems of Linear Equations: Three Variables

For the exercises 1-8, solve the system of three equations using substitution or addition.

1) \begin{align*} 0.5x-0.5y &= 10\\ -0.2y+0.2x &= 4\\ 0.1x+0.1z &= 2 \end{align*}

$$(10,-10,10)$$

2) \begin{align*} 5x+3y-z &= 5\\ 3x-2y+4z &= 13\\ 4x+3y+5z &= 22 \end{align*}

3) \begin{align*} x+y+z &= 1\\ 2x+2y+2z &= 1\\ 3x+3y &= 2 \end{align*}

No solutions exist.

4) \begin{align*} 2x-3y+z &= -1\\ x+y+z &= -4\\ 4x+2y-3z &= 33 \end{align*}

5) \begin{align*} 3x+2y-z &= -10\\ x-y+2z &= 7\\ -x+3y+z &= -2 \end{align*}

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

6) \begin{align*} 3x+4z &= -11\\ x-2y &= 5\\ 4y-z &= -10 \end{align*}

7) \begin{align*} 2x-3y+z &= 0\\ 2x+4y-3z &= 0\\ 6x-2y-z &= 0 \end{align*}

$$\left (x, \dfrac{8x}{5}, \dfrac{14x}{5} \right )$$

8) \begin{align*} 6x-4y-2z &= 2\\ 3x+2y-5z &= 4\\ 6y-7z &= 5 \end{align*}

For the exercises 9-10, write a system of equations to solve each problem. Solve the system of equations.

9) Three odd numbers sum up to $$61$$. The smaller is one-third the larger and the middle number is $$16$$ less than the larger. What are the three numbers?

$$11, 17, 33$$

10) A local theatre sells out for their show. They sell all $$500$$ tickets for a total purse of $$\8,070.00$$. The tickets were priced at $$\15$$ for students, $$\12$$ for children, and $$\18$$ for adults. If the band sold three times as many adult tickets as children’s tickets, how many of each type was sold?

## 9.3: Systems of Nonlinear Equations and Inequalities: Two Variables

For the exercises 1-5, solve the system of nonlinear equations.

1) \begin{align*} y &= x^2 - 7\\ y &= 5x-13 \end{align*}

$$(2,−3),(3,2)$$

2) \begin{align*} y &= x^2 - 4\\ y &= 5x+10 \end{align*}

3) \begin{align*} x^2 + y^2 &= 16\\ y &= x-8 \end{align*}

No solution

4) \begin{align*} x^2 + y^2 &= 25\\ y &= x^2 + 5 \end{align*}

5) \begin{align*} x^2 + y^2 &= 4\\ y - x^2 &= 3 \end{align*}

No solution

For the exercises 6-7, graph the inequality.

6) $$y>x^2 - 1$$

7) $$\dfrac{1}{4}x^2 + y^2 < 4$$ For the exercises 8-10, graph the system of inequalities.

8) \begin{align*} x^2 + y^2 +2x &<3 \\ y &>-x^2 - 3 \end{align*}

9) \begin{align*} x^2 -2x + y^2 - 4x &< 4\\ y &<-x+4 \end{align*} 10) \begin{align*} x^2 + y^2 &< 1\\ y^2 &< x \end{align*}

## 9.4: Partial Fractions

For the exercises 1-8, decompose into partial fractions.

1) $$\dfrac{-2x+6}{x^2 +3x+2}$$

$$\dfrac{2}{x+2}, \dfrac{-4}{x+1}$$

2) $$\dfrac{10x+2}{4x^2 +4x+1}$$

3) $$\dfrac{7x+20}{x^2 +10x+25}$$

$$\dfrac{7}{x+5}, \dfrac{-15}{(x+5)^2}$$

4) $$\dfrac{x-18}{x^2 -12x+36}$$

5) $$\dfrac{-x^2 +36x + 70}{x^3 -125}$$

$$\dfrac{3}{x-5}, \dfrac{-4x+1}{x^2 +5x+25}$$

6) $$\dfrac{-5x^2 +6x-2}{x^3 +27}$$

7) $$\dfrac{x^3 -4x^2 +3x+11}{(x^2 -2)^2}$$

$$\dfrac{x-4}{(x^2 -2)}, \dfrac{5x+3}{(x^2 -2)^2}$$

8) $$\dfrac{4x^4 -2x^3 +22x^2 -6x+48}{x(x^2 +4)^2}$$

## 9.5: Matrices and Matrix Operations

For the exercises 1-12, perform the requested operations on the given matrices.

$A=\begin{bmatrix} 4 & -2\\ 1 & 3 \end{bmatrix}, \begin{bmatrix} 6 & 7 & -3\\ 11 & -2 & 4 \end{bmatrix}, C=\begin{bmatrix} 6 & 7\\ 11 & -2\\ 14 & 0 \end{bmatrix} D=\begin{bmatrix} 1 & -4 & 9\\ 10 & 5 & -7\\ 2 & 8 & 5 \end{bmatrix} E=\begin{bmatrix} 7 & -14 & 3\\ 2 & -1 & 3\\ 0 & 1 & 9 \end{bmatrix} \nonumber$

1) $$-4A$$

$$\begin{bmatrix} -16 & 8\\ -4 & -12 \end{bmatrix}$$

2) $$10D-6E$$

3) $$B+C$$

undefined; dimensions do not match

4) $$AB$$

5) $$BA$$

undefined; inner dimensions do not match

6) $$BC$$

7) $$CB$$

$$\begin{bmatrix} 113 & 28 & 10\\ 44 & 81 & -41\\ 84 & 98 & -42 \end{bmatrix}$$

8) $$DE$$

9) $$ED$$

$$\begin{bmatrix} -127 & -74 & 176\\ -2 & 11 & 40\\ 28 & 77 & 38 \end{bmatrix}$$

10) $$EC$$

11) $$CE$$

undefined; inner dimensions do not match

12) $$A^3$$

## 9.6: Solving Systems with Gaussian Elimination

For the exercises 1-2, write the system of linear equations from the augmented matrix. Indicate whether there will be a unique solution.

1) $$\left [ \begin{array}{ccc|c} 1 & 0 & -3 & 7 \\ 0 & 1 & 2 & -5\\ 0 & 0 & 0 & 0\\ \end{array} \right ]$$

\begin{align*} x-3z &= 7\\ y+2z &= -5 \end{align*}\; \; \text{with infinite solutions}

2) $$\left [ \begin{array}{ccc|c} 1 & 0 & 5 & -9 \\ 0 & 1 & -2 & 4\\ 0 & 0 & 0 & 3\\ \end{array} \right ]$$

For the exercises 3-5, write the augmented matrix from the system of linear equations.

3) \begin{align*} -2x+2y+z &= 7\\ 2x-8y+5z &= 0\\ 19x-10y+22z &= 3 \end{align*}

$$\left [ \begin{array}{ccc|c} -2 & 2 & 1 & 7 \\ 2 & -8 & 5 & 0\\ 19 & -10 & 22 & 3\\ \end{array} \right ]$$

4) \begin{align*} 4x+2y-3z &= 14\\ -12x+3y+z &= 100\\ 9x-6y+2z &= 31 \end{align*}

5) \begin{align*} x+3z &= 12\\ -x+4y &= 0\\ y+2z &= -7 \end{align*}

$$\left [ \begin{array}{ccc|c} 1 & 0 & 3 & 12 \\ -1 & 4 & 0 & 0\\ 0 & 1 & 2 & -7\\ \end{array} \right ]$$

For the exercises 6-10, solve the system of linear equations using Gaussian elimination.

6) \begin{align*} 3x-4y &= -7\\ -6x+8y &= 14 \end{align*}

7) \begin{align*} 3x-4y &= 1\\ -6x+8y &= 6 \end{align*}

No solutions exist.

8) \begin{align*} -1.1x-2.3y &= 6.2\\ -5.2x-4.1y &= 4.3 \end{align*}

9) \begin{align*} 2x+3y+2z &= 1\\ -4x-6y-4z &= -2\\ 10x+15y+10z &= 0 \end{align*}

No solutions exist.

10) \begin{align*} -x+2y-4z &= 8\\ 3y+8z &= -4\\ -7x+y+2z &= 1 \end{align*}

## 9.7: Solving Systems with Inverses

For the exercises 1-4, find the inverse of the matrix.

1) $$\begin{bmatrix} -0.2 & 1.4\\ 1.2 & -0.4 \end{bmatrix}$$

$$\dfrac{1}{8}\begin{bmatrix} 2 & 7\\ 6 & 1 \end{bmatrix}$$

2) $$\begin{bmatrix} \frac{1}{2} & -\frac{1}{2}\\ -\frac{1}{4} & \frac{3}{4} \end{bmatrix}$$

3) $$\begin{bmatrix} 12 & 9 & -6\\ -1 & 3 & 2\\ -4 & -3 & 2 \end{bmatrix}$$

No inverse exists.

4) $$\begin{bmatrix} 2 & 1 & 3\\ 1 & 2 & 3\\ 3 & 2 & 1 \end{bmatrix}$$

For the exercises 5-8, find the solutions by computing the inverse of the matrix.

5) \begin{align*} 0.3x-0.1y &= -10\\ -0.1x+0.3y &= 14 \end{align*}

$$(-20,40)$$

6) \begin{align*} 0.4x-0.2y &= -0.6\\ -0.1x+0.05y &= 0.3 \end{align*}

7) \begin{align*} 4x+3y-3z &= -4.3\\ 5x-4y-z &= -6.1\\ x+z &= -0.7 \end{align*}

$$(-1, 0.2, 0.3)$$

8) \begin{align*} -2x-3y+2z &= 3\\ -x+2y+4z &= -5\\ -2y+5z &= -3 \end{align*}

For the exercises 9-10, write a system of equations to solve each problem. Solve the system of equations.

9) Students were asked to bring their favorite fruit to class. $$90\%$$ of the fruits consisted of banana, apple, and oranges. If oranges were half as popular as bananas and apples were $$5\%$$ more popular than bananas, what are the percentages of each individual fruit?

$$17\%$$ oranges, $$34\%$$ bananas, $$39\%$$ apples

10) A sorority held a bake sale to raise money and sold brownies and chocolate chip cookies. They priced the brownies at $$\2$$ and the chocolate chip cookies at $$\1$$. They raised $$\250$$ and sold $$175$$ items. How many brownies and how many cookies were sold?

## 9.8: Solving Systems with Cramer's Rule

For the exercises 1-4, find the determinant.

1) $$\begin{vmatrix} 100 & 0\\ 0 & 0 \end{vmatrix}$$

$$0$$

2) $$\begin{vmatrix} 0.2 & -0.6\\ 0.7 & -1.1 \end{vmatrix}$$

3) $$\begin{vmatrix} -1 & 4 & 3\\ 0 & 2 & 3\\ 0 & 0 & -3 \end{vmatrix}$$

$$6$$

4) $$\begin{vmatrix} \sqrt{2} & 0 & 0\\ 0 & \sqrt{2} & 0\\ 0 & 0 & \sqrt{2} \end{vmatrix}$$

For the exercises 5-10, use Cramer’s Rule to solve the linear systems of equations.

5) \begin{align*} 4x-2y &= 23\\ -5x-10y &= -35 \end{align*}

$$\left(6, \dfrac{1}{2} \right)$$

6) \begin{align*} 0.2x-0.1y &= 0\\ -0.3x+0.3y &= 2.5 \end{align*}

7) \begin{align*} -0.5x+0.1y &= 0.3\\ -0.25x+0.05y &= 0.15 \end{align*}

$$x, 5x+3$$

8) \begin{align*} x+6y+3z &= 4\\ 2x+y+2z &= 3\\ 3x-2y+z &= 0 \end{align*}

9) \begin{align*} 4x-3y+5z &= -\dfrac{5}{2}\\ 7x-9y-3z &= \dfrac{3}{2}\\ x-5y-5z &= \dfrac{5}{2} \end{align*}

$$\left(0, 0, -\dfrac{1}{2} \right)$$

10) \begin{align*} \dfrac{3}{10}x-\dfrac{1}{5}y-\dfrac{3}{10}z &= -\dfrac{1}{50}\\ \dfrac{1}{10}x-\dfrac{1}{10}y-\dfrac{1}{2}z &= -\dfrac{9}{50}\\ \dfrac{2}{5}x-\dfrac{1}{2}y-\dfrac{3}{5}z &= -\dfrac{1}{5} \end{align*}

## Practice Test

1) Is the following ordered pair a solution to the system of equations? \begin{align*} -5x-y &= 12 \text{ with } (-3,3)\\ x+4y &= 9 \end{align*} \nonumber

Yes

For the exercises 2-9, solve the systems of linear and nonlinear equations using substitution or elimination. Indicate if no solution exists.

2) \begin{align*} \dfrac{1}{2}x-\dfrac{1}{3}y &= 4\\ \dfrac{3}{2}x-y &= 0 \end{align*}

3) \begin{align*} -\dfrac{1}{2}x-4y &= 4\\ 2x+16y &= 2 \end{align*}

No solutions exist.

4) \begin{align*} 5x-y &= 1\\ -10x+2y &= -2 \end{align*}

5) \begin{align*} 4x-6y-2z &= \dfrac{1}{10}\\ x-7y+5z &= -\dfrac{1}{4}\\ 3x+6y-9z &= \dfrac{6}{5} \end{align*}

$$\dfrac{1}{20} (10, 5, 4)$$

6) \begin{align*} x+z &= 20\\ x+y+z &= 20\\ x+2y+z &= 10 \end{align*}

7) \begin{align*} 5x-4y-3z &= 0\\ 2x+y+2z &= 0\\ x-6y-7z &= 0 \end{align*}

$$\left ( x, \dfrac{16x}{5} - \dfrac{13x}{5} \right )$$

8) \begin{align*} y &= x^2 +2x-3\\ y &= x-1 \end{align*}

9) \begin{align*} y^2 + x^2 &= 25\\ y^2 -2x^2 &= 1 \end{align*}

$$(-2\sqrt{2}, -\sqrt{17}), (-2\sqrt{2}, \sqrt{17}), (2\sqrt{2}, -\sqrt{17}), (2\sqrt{2}, \sqrt{17})$$

For the exercises 10-11, graph the following inequalities.

10) $$y < x^2 + 9$$

11) \begin{align*} x^2 + y^2 &> 4 \\ y &< x^2 + 1 \end{align*} For the exercises 12-14, write the partial fraction decomposition.

12) $$\dfrac{-8x-30}{x^2 + 10x+25}$$

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

$$\dfrac{5}{3x+1}-\dfrac{2x+3}{(3x+1)^2}$$

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

For the exercises 15-21, perform the given matrix operations.

15) $$5\begin{bmatrix} 4 & 9\\ -2 & 3 \end{bmatrix}+\dfrac{1}{2} \begin{bmatrix} -6 & 12\\ 4 & -8 \end{bmatrix}$$

$$\begin{bmatrix} 17 & 51\\ -8 & 11 \end{bmatrix}$$

16) $$\begin{bmatrix} 1 & 4 & -7\\ -2 & 9 & 5\\ 12 & 0 & -4 \end{bmatrix} \begin{bmatrix} 3 & -4\\ 1 & 3\\ 5 & 10 \end{bmatrix}$$

17) $$\begin{bmatrix} \frac{1}{2} & \frac{1}{3}\\ \frac{1}{4} & \frac{1}{5} \end{bmatrix} ^{-1}$$

$$\begin{bmatrix} 12 & -20\\ -15 & 30 \end{bmatrix}$$

18) $$\textbf{det}\begin{vmatrix} 0 & 0\\ 400 & 4,000 \end{vmatrix}$$

19) $$\textbf{det}\begin{vmatrix} \frac{1}{2} & -\frac{1}{2} & 0\\ -\frac{1}{2} & 0 & \frac{1}{2}\\ 0 & \frac{1}{2} & 0 \end{vmatrix}$$

$$-\dfrac{1}{8}$$

20) If $$\textbf{det}(A)=-6$$, 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{align*} 14x-2y-13z &= 140\\ -2x+3y-6z &= -1\\ x-5y+12z &= 11 \end{align*} \nonumber

$$\left [ \begin{array}{ccc|c} 14 & -2 & 13 & 140 \\ -2 & 3 & -6 & -1\\ 1 & -5 & 12 & 11\\ \end{array} \right ]$$

22) Rewrite the augmented matrix as a system of linear equations. $\left [ \begin{array}{ccc|c} 1 & 0 & 3 & 12 \\ -2 & 4 & 9 & -5\\ -6 & 1 & 2 & 8\\ \end{array} \right ] \nonumber$

For the exercises 23-24, use Gaussian elimination to solve the systems of equations.

23) \begin{align*} x-6y &= 4\\ 2x-12y &= 0 \end{align*}

No solutions exist.

24) \begin{align*} 2x+y+z &= -3\\ x-2y+3z &= 6\\ x-y-z &= 6 \end{align*}

For the exercises 25-26, use the inverse of a matrix to solve the systems of equations.

25) \begin{align*} 4x-5y &= -50\\ -x+2y &= 80 \end{align*}

$$(100, 90)$$

26) \begin{align*} \dfrac{1}{100}x-\dfrac{3}{100}y+\dfrac{1}{20}z &= -49\\ \dfrac{3}{100}x-\dfrac{7}{100}y-\dfrac{1}{100}z &= 13\\ \dfrac{9}{100}x-\dfrac{9}{100}y-\dfrac{9}{100}z &= 99 \end{align*}

For the exercises 27-28, use Cramer’s Rule to solve the systems of equations.

27) \begin{align*} 200x-300y &= 2\\ 400x+715y &= 4 \end{align*}

$$\left (\dfrac{1}{100}, 0 \right )$$

28) \begin{align*} 0.1x+0.1y-0.1z &= -1.2\\ 0.1x-0.2y+0.4z &= -1.2\\ 0.5x-0.3y+0.8z &= -5.9 \end{align*}

For the exercises 29-30, solve using a system of linear equations.

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

$$32$$ or more cell phones per day
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?