Chapter 8

In Exercises \(7-16\), sketch the graph of the system of linear inequalities. $$ \left\\{\begin{array}{l} y \leq x-5 \\ y>-7 \end{array}\right. $$

In Exercises \(11-16\), use a system of linear equations to find the dimensions of the rectangle that meet the specified conditions. 40 feet The length is 4 feet greater than the width.

In Exercises 7-12, solve the system by the method of elimination. $$ \left\\{\begin{array}{l} 2 x-5 y=-1 \\ 2 x-y=1 \end{array}\right. $$

In Exercises 5-14, solve the system by the method of substitution. $$ \left\\{\begin{array}{l} 2 x-y=-2 \\ 4 x+y=5 \end{array}\right. $$

In Exercises \(7-16\), sketch the graph of the system of linear inequalities. $$ \left\\{\begin{array}{l} y<-x+3 \\ y \geq-1 \end{array}\right. $$

In Exercises \(11-16\), use a system of linear equations to find the dimensions of the rectangle that meet the specified conditions. 220 inches The width is 10 inches less than the length.

In Exercises 7-12, solve the system by the method of elimination. $$ \left\\{\begin{array}{l} 7 x+8 y=6 \\ 3 x-4 y=10 \end{array}\right. $$

In Exercises 5-14, solve the system by the method of substitution. $$ \left\\{\begin{array}{rr} x+6 y & =7 \\ -x+4 y & =-2 \end{array}\right. $$

In Exercises \(7-16\), sketch the graph of the system of linear inequalities. $$ \left\\{\begin{array}{l} y>2 x \\ y>-x+4 \end{array}\right. $$

In Exercises \(11-16\), use a system of linear equations to find the dimensions of the rectangle that meet the specified conditions. 16 yards The width is one-third of the length.

In Exercises \(13-18\), solve the system by the method of elimination. $$ \left\\{\begin{array}{l} 4 x+5 y=7 \\ 6 x-2 y=-18 \end{array}\right. $$

In Exercises 5-14, solve the system by the method of substitution. $$ \left\\{\begin{aligned} x+2 y &=1 \\ 5 x-4 y &=-23 \end{aligned}\right. $$

In Exercises \(7-16\), sketch the graph of the system of linear inequalities. $$ \left\\{\begin{array}{l} y \leq-x \\ y \leq x+1 \end{array}\right. $$

In Exercises \(13-18\), solve the system by the method of elimination. $$ \left\\{\begin{array}{l} 5 x+3 y=18 \\ 2 x-7 y=-1 \end{array}\right. $$

In Exercises 5-14, solve the system by the method of substitution. $$ \left\\{\begin{aligned} -2 x+y &=-18 \\ 3 x+3 y &=9 \end{aligned}\right. $$

In Exercises \(7-16\), sketch the graph of the system of linear inequalities. $$ \left\\{\begin{array}{l} x+y>-1 \\ x+y<3 \end{array}\right. $$

In Exercises \(11-16\), use a system of linear equations to find the dimensions of the rectangle that meet the specified conditions. \(35.2\) meters The length is \(120 \%\) of the width.

In Exercises \(13-18\), solve the system by the method of elimination. $$ \left\\{\begin{array}{l} 3 x+2 y=10 \\ 2 x+5 y=3 \end{array}\right. $$

In Exercises \(15-20\), solve the system by the method of substitution. $$ \left\\{\begin{array}{r} 8 x+4 y=-2 \\ -12 x+5 y=-8 \end{array}\right. $$

In Exercises \(7-16\), sketch the graph of the system of linear inequalities. $$ \left\\{\begin{array}{l} x-y>2 \\ x-y<-4 \end{array}\right. $$

In Exercises \(11-16\), use a system of linear equations to find the dimensions of the rectangle that meet the specified conditions. 35 feet The width is \(75 \%\) of the length.

In Exercises \(13-18\), solve the system by the method of elimination. $$ \left\\{\begin{array}{l} 5 u+6 v=14 \\ 3 u+5 v=7 \end{array}\right. $$

In Exercises \(15-20\), solve the system by the method of substitution. $$ \left\\{\begin{array}{l} 5 x-2 y=10 \\ 3 x+2 y=6 \end{array}\right. $$

In Exercises 17-22, sketch the graph of the system of linear inequalities, and label the vertices. $$ \left\\{\begin{array}{lr} y> & -5 \\ x \leq & 2 \\ y \leq x+2 \end{array}\right. $$

In Exercises \(13-18\), solve the system by the method of elimination. $$ \left\\{\begin{array}{l} 2 x+3 y=16 \\ 5 x-10 y=30 \end{array}\right. $$

In Exercises \(15-20\), solve the system by the method of substitution. $$ \left\\{\begin{array}{l} 3 x+7 y=2 \\ 5 x-3 y=-26 \end{array}\right. $$

In Exercises 17-22, sketch the graph of the system of linear inequalities, and label the vertices. $$ \left\\{\begin{array}{lr} y \geq & -1 \\ x \leq & 5 \\ y \leq x+4 \end{array}\right. $$

You travel 16 miles upstream and 16 miles downstream on a motorboat trip. You run the motor at the same speed going up and down the river, but because of the speed of the current, the trip upstream takes 2 hours and the trip downstream take \(1 \frac{1}{3}\) hours. Determine the speed of the boat in still water.

In Exercises \(13-18\), solve the system by the method of elimination. $$ \left\\{\begin{array}{l} 3 x-4 y=1 \\ 4 x+3 y=1 \end{array}\right. $$

In Exercises \(15-20\), solve the system by the method of substitution. $$ \left\\{\begin{array}{l} 4 x+3 y=15 \\ 2 x-5 y=1 \end{array}\right. $$

List the three methods for solving systems of linear equations.

In Exercises \(19-24\), solve the system by the method of elimination. $$ \left\\{\begin{array}{l} 0.02 x-0.05 y=-0.19 \\ 0.03 x+0.04 y=0.52 \end{array}\right. $$

In Exercises \(15-20\), solve the system by the method of substitution. $$ \left\\{\begin{array}{l} 2 x-3 y=16 \\ 3 x+4 y=7 \end{array}\right. $$

In Exercises 19-26, solve the system by graphing. $$ \left\\{\begin{array}{r} x+y=4 \\ 4 x+4 y=2 \end{array}\right. $$

In Exercises 17-22, sketch the graph of the system of linear inequalities, and label the vertices. $$ \left\\{\begin{aligned} 2 x+y & \geq 2 \\ x & \leq 2 \\ y & \leq 1 \end{aligned}\right. $$

How can you determine whether a real-life problem may be modeled with a system of linear equations?

In Exercises \(19-24\), solve the system by the method of elimination. $$ \left\\{\begin{array}{l} 0.05 x-0.03 y=0.21 \\ 0.01 x+0.01 y=0.09 \end{array}\right. $$

In Exercises \(15-20\), solve the system by the method of substitution. $$ \left\\{\begin{aligned} -2 x+7 y &=9 \\ 3 x+2 y &=-1 \end{aligned}\right. $$

In Exercises 19-26, solve the system by graphing. $$ \left\\{\begin{array}{r} 3 x+3 y=1 \\ x+y=3 \end{array}\right. $$

In Exercises 17-22, sketch the graph of the system of linear inequalities, and label the vertices. $$ \left\\{\begin{array}{r} x+y \leq 1 \\ -x+y \leq 1 \\ y \geq 0 \end{array}\right. $$

What is a verbal model of a real-life problem?

In Exercises \(19-24\), solve the system by the method of elimination. $$ \left\\{\begin{array}{l} 0.1 x-0.1 y=0 \\ 0.8 x+0.3 y=1.5 \end{array}\right. $$

In Exercises 21-26, solve the system by the method of substitution. $$ \left\\{\begin{array}{l} 4 x-y=2 \\ 2 x-\frac{1}{2} y=1 \end{array}\right. $$

In Exercises 19-26, solve the system by graphing. $$ \left\\{\begin{array}{l} x+2 y=3 \\ x-3 y=13 \end{array}\right. $$

In Exercises 17-22, sketch the graph of the system of linear inequalities, and label the vertices. $$ \left\\{\begin{aligned} 3 x+2 y &<6 \\ -3 x+2 y & \geq 6 \\ y & \geq 0 \end{aligned}\right. $$

Compare the one-variable method with the two-variable method for modeling real-life problems.

In Exercises \(19-24\), solve the system by the method of elimination. $$ \left\\{\begin{aligned} x-2 y &=0 \\ 0.2 x+0.8 y &=2.4 \end{aligned}\right. $$

In Exercises 21-26, solve the system by the method of substitution. $$ \left\\{\begin{array}{r} x-5 y=5 \\ 3 x-15 y=15 \end{array}\right. $$

In Exercises 19-26, solve the system by graphing. $$ \left\\{\begin{array}{r} -x+10 y=30 \\ x+10 y=10 \end{array}\right. $$

In Exercises 23-28, sketch the graph of the system of linear inequalities. $$ \left\\{\begin{array}{r} -3 x+2 y<6 \\ x-4 y>-2 \\ 2 x+y<3 \end{array}\right. $$