Given the cost function $C \left(x\right) = 0.85 x + 35,000$ and the revenue function $R \left(x\right) = 1.55 x ,$ find the break-even point and the profit function.

2.5

Writing and Solving a System of Equations in Two Variables

The cost of a ticket to the circus is $\$ 25.00$ for children and $\$ 50.00$ for adults. On a certain day, attendance at the circus is $2,000$ and the total gate revenue is $\$ 70,000.$ How many children and how many adults bought tickets?

2.5

11.1 Exercise 25

Meal tickets at the circus cost $\$ 4.00$ for children and $\$ 12.00$ for adults. If $1,650$ meal tickets were bought for a total of $\$ 14,200 ,$ how many children and how many adults bought meal tickets?

2.5

11.1 Exercise 26

Can a system of linear equations have exactly two solutions? Explain why or why not.

2.5

11.1 Exercise 28

If you are solving a break-even analysis and get a negative break-even point, explain what this signifies for the company?

2.5

11.1 Exercise 30

Given a system of equations, explain at least two different methods of solving that system.

2.5

11.1 Exercise 32

Determine whether the given ordered pair is a solution to the system of equations.

$\begin{matrix}−3 x - 5 y = 13 \\ - x + 4 y = 10\end{matrix}$ and $\left(−6 , 1\right)$

2.5

11.1 Exercise 34

Determine whether the given ordered pair is a solution to the system of equations.

$\begin{matrix}−2 x + 5 y = 7 \\ 2 x + 9 y = 7\end{matrix}$ and $\left(−1 , 1\right)$

2.5

11.1 Exercise 36

Solve each system by substitution.

$\begin{matrix} x + 3 y = 5 \\ 2 x + 3 y = 4\end{matrix}$

2.5

11.1 Exercise 38

Solve each system by substitution.

$\begin{matrix}4 x + 2 y = −10 \\ 3 x + 9 y = 0\end{matrix}$

2.5

11.1 Exercise 40

Solve each system by substitution.

$\begin{matrix}- 2 x + 3 y = 1.2 \\ - 3 x - 6 y = 1.8\end{matrix}$

2.5

11.1 Exercise 42

Solve each system by substitution.

$\begin{matrix} \text{ } 3 x + 5 y = 9 \\ 30 x + 50 y = −90\end{matrix}$

2.5

11.1 Exercise 44

Solve each system by substitution.

$\begin{matrix}\frac{1}{2} x + \frac{1}{3} y = 16 \\ \frac{1}{6} x + \frac{1}{4} y = 9\end{matrix}$

2.5

11.1 Exercise 46

Solve each system by addition.

$\begin{matrix}−2 x + 5 y = −42 \\ 7 x + 2 y = 30\end{matrix}$

2.5

11.1 Exercise 48

Solve each system by addition.

$\begin{matrix} \text{ } 5 x - y = −2.6 \\ −4 x −6 y = 1.4\end{matrix}$

2.5

11.1 Exercise 50

Solve each system by addition.

$\begin{matrix} −x + 2 y = −1 \\ 5 x −10 y = 6\end{matrix}$

2.5

11.1 Exercise 52

Solve each system by addition.

$\begin{matrix}\frac{5}{6} x + \frac{1}{4} y = 0 \\ \frac{1}{8} x - \frac{1}{2} y = - \frac{43}{120}\end{matrix}$

2.5

11.1 Exercise 54

Solve each system by addition.

$\begin{matrix} \\ \begin{matrix}−0.2 x + 0.4 y = 0.6 \\ \text{ } x −2 y = −3\end{matrix}\end{matrix}$

2.5

11.1 Exercise 56

Solve each system by any method.

$\begin{matrix}5 x + 9 y = 16 \\ \text{ } x + 2 y = 4\end{matrix}$

2.5

11.1 Exercise 58

Solve each system by any method.

$\begin{matrix}5 x −2 y = 2.25 \\ 7 x −4 y = 3\end{matrix}$

2.5

11.1 Exercise 60

Solve each system by any method.

$\begin{matrix}7 x −4 y = \frac{7}{6} \\ 2 x + 4 y = \frac{1}{3}\end{matrix}$

2.5

11.1 Exercise 62

Solve each system by any method.

$\begin{matrix} \text{ } \frac{7}{3} x - \frac{1}{6} y = 2 \\ - \frac{21}{6} x + \frac{3}{12} y = −3\end{matrix}$

2.5

11.1 Exercise 64

Solve each system by any method.

$\begin{matrix}2.2 x + 1.3 y = −0.1 \\ 4.2 x + 4.2 y = 2.1\end{matrix}$

2.5

11.1 Exercise 66

Graph the system of equations and state whether the system is consistent, inconsistent, or dependent and whether the system has one solution, no solution, or infinite solutions.

$\begin{matrix}3 x - y = 0.6 \\ x −2 y = 1.3\end{matrix}$

2.5

11.1 Exercise 68

Graph the system of equations and state whether the system is consistent, inconsistent, or dependent and whether the system has one solution, no solution, or infinite solutions.

$\begin{matrix} x + 2 y = 7 \\ 2 x + 6 y = 12\end{matrix}$

2.5

11.1 Exercise 70

Graph the system of equations and state whether the system is consistent, inconsistent, or dependent and whether the system has one solution, no solution, or infinite solutions.

$\begin{matrix} 3 x −2 y = 5 \\ −9 x + 6 y = −15\end{matrix}$

2.5

11.1 Exercise 72

Use the intersect function on a graphing device to solve each system. Round all answers to the nearest hundredth.

$\begin{matrix}−0.01 x + 0.12 y = 0.62 \\ 0.15 x + 0.20 y = 0.52\end{matrix}$

2.5

11.1 Exercise 74

Use the intersect function on a graphing device to solve each system. Round all answers to the nearest hundredth.

$\begin{matrix} 0.15 x + 0.27 y = 0.39 \\ −0.34 x + 0.56 y = 1.8\end{matrix}$

2.5

11.1 Exercise 76

Solve each system in terms of $A , B , C , D , E ,$ and $F$ where $A – F$ are nonzero numbers. Note that $A \neq B$ and $A E \neq B D .$

$\begin{matrix}x + y = A \\ x - y = B\end{matrix}$

2.5

11.1 Exercise 78

Solve each system in terms of $A , B , C , D , E ,$ and $F$ where $A – F$ are nonzero numbers. Note that $A \neq B$ and $A E \neq B D .$

$\begin{matrix}A x + y = 0 \\ B x + y = 1\end{matrix}$

2.5

11.1 Exercise 80

Solve each system in terms of $A , B , C , D , E ,$ and $F$ where $A – F$ are nonzero numbers. Note that $A \neq B$ and $A E \neq B D .$

$\begin{matrix}A x + B y = C \\ D x + E y = F\end{matrix}$

2.5

11.1 Exercise 82

Solve for the desired quantity.

An Ethiopian restaurant has a cost of production $C \left(x\right) = 11 x + 120$ and a revenue function $R \left(x\right) = 5 x .$ When does the company start to turn a profit?

2.5

11.1 Exercise 84

Solve for the desired quantity.

A musician charges $C \left(x\right) = 64 x + 20,000$ where $x$ is the total number of attendees at the concert. The venue charges $80 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?

2.5