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


## 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)

Answer

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*}

Answer

$$(-2,3)$$

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*}

Answer

$$(4,-1)$$

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*}

Answer

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?

Answer

$$(300,60,000)$$

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?

Answer

$$(400,30,000)$$

## 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*}

Answer

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

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*}

Answer

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*}

Answer

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

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*}

Answer

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

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?

Answer

$$11, 17, 33$$

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*}

Answer