Chapter 8

In Exercises 27-36, solve the system by graphing. $$ \left\\{\begin{array}{l} 7 x+3 y=21 \\ \frac{7}{3} x+y=7 \end{array}\right. $$

In Exercises 35-38, use a graphing calculator to graph the cost and revenue equations in the same viewing window. Find the sales \(x\) necessary to break even \((R=C)\) and the corresponding revenue \(R\) obtained by selling \(x\) units. (Round \(x\) to the nearest whole unit.) $$ C=7650 x+125,000 \quad R=8950 x $$

In Exercises 35-38, solve the system by the method of elimination. $$ \left\\{\begin{array}{r} -\frac{x}{4}+y=1 \\ \frac{x}{4}+\frac{y}{2}=1 \end{array}\right. $$

In Exercises 35-46, solve the system by the method of substitution. $$ \left\\{\begin{array}{l} y=\frac{1}{4} x+\frac{19}{4} \\ y=\frac{8}{5} x-2 \end{array}\right. $$

In Exercises 27-36, solve the system by graphing. $$ \left\\{\begin{array}{r} 8 x-6 y=-12 \\ x-\frac{3}{4} y=-2 \end{array}\right. $$

In Exercises 35-38, use a graphing calculator to graph the cost and revenue equations in the same viewing window. Find the sales \(x\) necessary to break even \((R=C)\) and the corresponding revenue \(R\) obtained by selling \(x\) units. (Round \(x\) to the nearest whole unit.) $$ C=2175 x+85,000 \quad R=3525 x $$

In Exercises 35-38, solve the system by the method of elimination. $$ \left\\{\begin{array}{c} \frac{x}{3}-\frac{y}{5}=1 \\ \frac{x}{12}+\frac{y}{40}=1 \end{array}\right. $$

In Exercises 35-46, solve the system by the method of substitution. $$ \left\\{\begin{array}{l} y=\frac{5}{4} x+3 \\ y=\frac{1}{2} x+6 \end{array}\right. $$

In Exercises 27-36, solve the system by graphing. $$ \left\\{\begin{array}{r} -x+\frac{2}{3} y=5 \\ 9 x-6 y=6 \end{array}\right. $$

When should the line that corresponds to an inequality be dashed? When should it be solid?

In Exercises 35-38, use a graphing calculator to graph the cost and revenue equations in the same viewing window. Find the sales \(x\) necessary to break even \((R=C)\) and the corresponding revenue \(R\) obtained by selling \(x\) units. (Round \(x\) to the nearest whole unit.) $$ C=0.55 x+40,000 \quad R=0.85 x $$

In Exercises 35-38, solve the system by the method of elimination. $$ \left\\{\begin{aligned} 3(x+5)-7 &=2(3-2 y) \\ 2 x+1 &=4(y+2) \end{aligned}\right. $$

In Exercises 35-46, solve the system by the method of substitution. $$ \left\\{\begin{array}{r} 3 x+2 y=12 \\ x-y=3 \end{array}\right. $$

A small company produces bird feeders that sell for \(\$ 23\) per unit. The cost of producing each unit is \(\$ 16.75\), and the company has fixed costs of \(\$ 400\). (a) Use a verbal model to show that the cost \(C\) of producing \(x\) units is \(C=16.75 x+400\) and the revenue \(R\) from selling \(x\) units is \(R=23 x\). (b) Use a graphing calculator to graph the cost and revenue functions in the same viewing window. Approximate the point of intersection of the graphs and internret the result

How do you check a single point \(\left(x_{1}, y_{1}\right)\) to determine whether it is a solution of a system of inequalities?

In Exercises 35-38, use a graphing calculator to graph the cost and revenue equations in the same viewing window. Find the sales \(x\) necessary to break even \((R=C)\) and the corresponding revenue \(R\) obtained by selling \(x\) units. (Round \(x\) to the nearest whole unit.) $$ C=0.25 x+25,000 \quad R=0.45 x $$

In Exercises 35-38, solve the system by the method of elimination. $$ \left\\{\begin{aligned} \frac{1}{2}(x-4)+9 &=y-10 \\ -5(x+3) &=8-2(y-3) \end{aligned}\right. $$

In Exercises 35-46, solve the system by the method of substitution. $$ \left\\{\begin{array}{l} 16 x-8 y=5 \\ 32 x+8 y=19 \end{array}\right. $$

A company produces hockey sticks that sell for \(\$ 79\) per unit. The cost of producing each unit is \(\$ 53.25\), and the company has fixed costs of \(\$ 1000\). (a) Use a verbal model to show that the cost \(C\) of producing \(x\) units is \(C=53.25 x+1000\) and the revenue \(R\) from selling \(x\) units is \(R=79 x\). (b) Use a graphing calculator to graph the cost and revenue functions in the same viewing window. Approximate the point of intersection of the graphs and interpret the result.

Explain how to sketch the graph of a system of inequalities in two variables.

In Exercises 39-42, find \(m\) and \(b\) such that \(y=m x+b\) is the equation of the line through the points. $$ (2,-1),(6,1) $$

You invest a total of \(\$ 10,000\) in two funds earning \(7.5 \%\) and \(10 \%\) simple interest. (There is more risk in the \(10 \%\) fund.) Your goal is to have a total annual interest income of \(\$ 850\). The system of equations that represents this situation is $$ \left\\{\begin{aligned} x+y &=10,000 \\ 0.075 x+0.10 y &=850 \end{aligned}\right. $$ where \(x\) is the amount invested in the \(7.5 \%\) fund and \(y\) is the amount invested in the \(10 \%\) fund. Solve this system to determine the smallest amount that you can invest at \(10 \%\) in order to meet your objective.

In Exercises 35-46, solve the system by the method of substitution. $$ \left\\{\begin{array}{l} 5 x+3 y=15 \\ 2 x-3 y=6 \end{array}\right. $$

How do you determine the vertices of the solution region for a system of linear inequalities?

In Exercises 39-42, find \(m\) and \(b\) such that \(y=m x+b\) is the equation of the line through the points. $$ (1,3),(4,9) $$

You invest a total of \(\$ 12,000\) in two funds earning \(8 \%\) and \(11.5 \%\) simple interest. (There is more risk in the \(11.5 \%\) fund.) Your goal is to have a total annual interest income of \(\$ 1065\). The system of equations that represents this situation is $$ \left\\{\begin{aligned} x+y &=12,000 \\ 0.08 x+0.115 y &=1,065 \end{aligned}\right. $$ where \(x\) is the amount invested in the \(8 \%\) fund and \(y\) is the amount invested in the \(11.5 \%\) fund. Solve this system to determine the smallest amount that you can invest at \(11.5 \%\) in order to meet your objective.

In Exercises 35-46, solve the system by the method of substitution. $$ \left\\{\begin{array}{l} 4 x-5 y=0 \\ 2 x-5 y=-10 \end{array}\right. $$

What is a dependent system of linear equations?

In Exercises \(41-46\), sketch the graph of the system of linear inequalities. $$ \left\\{\begin{aligned} x+y & \leq 4 \\ x & \geq 0 \\ y & \geq 0 \end{aligned}\right. $$

In Exercises 39-42, find \(m\) and \(b\) such that \(y=m x+b\) is the equation of the line through the points. $$ (-3,6),(5,2) $$

In Exercises 41 and 42 , solve the system to find the two numbers. The sum of two numbers \(x\) and \(y\) is 82 and the difference of the numbers is 14 . The systems of equations that represents this problem is $$ \left\\{\begin{array}{l} x+y=82 \\ x-y=14 \end{array}\right. \text {. } $$

In Exercises 35-46, solve the system by the method of substitution. $$ \left\\{\begin{array}{l} \frac{x}{3}-\frac{y}{4}=2 \\ \frac{x}{2}+\frac{y}{6}=3 \end{array}\right. $$

What is an inconsistent system of linear equations?

In Exercises \(41-46\), sketch the graph of the system of linear inequalities. $$ \left\\{\begin{array}{r} 2 x+y \leq 6 \\ x \geq 0 \\ y \geq 0 \end{array}\right. $$

In Exercises 39-42, find \(m\) and \(b\) such that \(y=m x+b\) is the equation of the line through the points. $$ (0,2),(4,-8) $$

In Exercises 41 and 42 , solve the system to find the two numbers. The sum of two numbers \(x\) and \(y\) is 154 and the difference of the numbers is 38 . The system of equations that represents this problem is $$ \left\\{\begin{array}{l} x+y=154 \\ x-y=38 \end{array}\right. \text {. } $$

In Exercises 35-46, solve the system by the method of substitution. $$ \left\\{\begin{aligned} -\frac{x}{5}+\frac{y}{2} &=-3 \\ \frac{x}{4}-\frac{y}{4} &=0 \end{aligned}\right. $$

What is one of the drawbacks to solving a system of linear equations by graphing?

Consider the one-variable method and the two-variable method for solving Example 4. Which do you prefer? Explain.

Find an equation of the line of slope \(m=\frac{1}{3}\) passing through the intersection of the lines \(3 x+4 y=7 \quad\) and \(5 x-4 y=1\).

In Exercises 35-46, solve the system by the method of substitution. $$ \left\\{\begin{array}{l} \frac{x}{4}+\frac{y}{2}=1 \\ \frac{x}{2}-\frac{y}{3}=1 \end{array}\right. $$

In Exercises \(43-48\), write the equations of the lines in slope-intercept form. What can you conclude about the number of solutions of the system? $$ \left\\{\begin{array}{r} 2 x-3 y=-12 \\ -8 x+12 y=-12 \end{array}\right. $$

In Exercises \(41-46\), sketch the graph of the system of linear inequalities. $$ \left\\{\begin{array}{r} 2 x-6 y>6 \\ x \quad y \leq 0 \end{array}\right. $$

Find an equation of the line of slope \(m=-2\) passing through the intersection of the lines \(2 x+5 y=11\) and \(4 x-y=11\).

In Exercises 35-46, solve the system by the method of substitution. $$ \left\\{\begin{aligned} -\frac{x}{6}+\frac{y}{12} &=1 \\ \frac{x}{2}+\frac{y}{8} &=1 \end{aligned}\right. $$

In Exercises \(43-48\), write the equations of the lines in slope-intercept form. What can you conclude about the number of solutions of the system? $$ \left\\{\begin{aligned} -5 x+8 y &=8 \\ -5 x+8 y &=-28 \end{aligned}\right. $$

In Exercises \(41-46\), sketch the graph of the system of linear inequalities. $$ \left\\{\begin{array}{r} x \quad \geq 1 \\ x-2 y \leq 3 \\ 3 x+2 y \geq 9 \\ x+y \leq 6 \end{array}\right. $$

Describe and correct the error in writing the system of linear equations for the problem below. Do not solve the system. The sum of two numbers is 42 , and the larger number is 3 less than twice the smaller number. $$ \begin{aligned} x+y &=42 \\ 2 x-y &=-3 \end{aligned} $$

Explain how to "clear" a system of decimals. Give an example to justify your answer. (There are many correct answers.)

In Exercises 35-46, solve the system by the method of substitution. $$ \left\\{\begin{aligned} 2(x-5) &=y+2 \\ 3 x &=4(y+2) \end{aligned}\right. $$