Chapter 7

Choose a solution method to solve the linear system. Explain your choice, and then solve the system. $$ \begin{aligned} &x+y=0\\\ &3 x+2 y=1 \end{aligned} $$

Use linear combinations to solve the linear system. Then check your solution. \(9 m-3 n=20\) \(3 m+6 n=2\)

Estimate the solution of the linear system graphically. Then check the solution algebraically. $$ \begin{aligned} &y=-2 x+6\\\ &y=2 x+2 \end{aligned} $$

Use the substitution method to solve the linear system. $$ \begin{array}{r} {u-v=0} \\ {7 u+v=0} \end{array} $$

Graph the system of linear inequalities. $$ \begin{array}{r} {x-2 y<3} \\ {3 x+2 y>9} \\ {x+y<6} \end{array} $$

Explain how you can tell from the equations how many solutions the linear system has. Then solve the system. $$x-y=2 \quad \text { Equation } 1$$ $$4 x-4 y=8 \quad \text { Equation } 2$$

Choose a solution method to solve the linear system. Explain your choice, and then solve the system. $$ \begin{array}{r} {2 x-3 y=-7} \\ {3 x+y=-5} \end{array} $$

Use linear combinations to solve the linear system. Then check your solution. \(x-3 y=30\) \(3 y+x=12\)

Estimate the solution of the linear system graphically. Then check the solution algebraically. $$ \begin{array}{c} {5 x+6 y=54} \\ {-x+y=9} \end{array} $$

Use the substitution method to solve the linear system. $$ \begin{array}{c} {x-y=0} \\ {12 x-5 y=-21} \end{array} $$

Use the substitution method or linear combinations to solve the linear system and tell how many solutions the system has. Then describe the graph of the system. $$\begin{aligned} -7 x+7 y &=7 \\ 2 x-2 y &=-18 \end{aligned}$$

Choose a solution method to solve the linear system. Explain your choice, and then solve the system. $$ \begin{array}{r} {8 x+4 y=8} \\ {-2 x+3 y=12} \end{array} $$

Use linear combinations to solve the linear system. Then check your solution. \(3 b+2 c=46\) \(5 c+b=11\)

Use the substitution method to solve the linear system. $$ \begin{array}{c} {m+2 n=1} \\ {5 m+3 n=-23} \end{array} $$

Use the substitution method or linear combinations to solve the linear system and tell how many solutions the system has. Then describe the graph of the system. $$\begin{aligned} &4 x+4 y=-8\\\ &2 x+2 y=-4 \end{aligned}$$

Choose a solution method to solve the linear system. Explain your choice, and then solve the system. $$ \begin{array}{c} {x+2 y=1} \\ {5 x-4 y=-23} \end{array} $$

Use linear combinations to solve the linear system. Then check your solution. \(y=x-9\) \(x+8 y=0\)

AEROBICS CLASSES A fitness club offers an aerobics class in the morning and in the evening. Assuming that the number of people in each class can be represented by a linear function, use the information in the table below to predict when the number of people in each class will be the same. $$ \begin{array}{|c|c|c|} \hline \text { Class } & {\text { Current }} & {\text { Increase (people }} \\\ \underline{\phantom{xxx}} & {\text { attendance }} & {\text { per month) }} \\ \hline \text { Morning } & {40} & {2} \\ \hline \text { Evening } & {22} & {8} \\ \hline \end{array} $$

Use the substitution method to solve the linear system. $$ \begin{aligned} &x-y=-5\\\ &x+4=16 \end{aligned} $$

Use the substitution method or linear combinations to solve the linear system and tell how many solutions the system has. Then describe the graph of the system. $$\begin{array}{c} {2 x+y=-4} \\ {4 x-2 y=8} \end{array}$$

Choose a solution method to solve the linear system. Explain your choice, and then solve the system. $$ \begin{aligned} &6 x-y=18\\\ &8 x+y=24 \end{aligned} $$

Use linear combinations to solve the linear system. Then check your solution. \(m=3 n\) \(m+10 n=13\)

The fast-changing world of the 1920 s produced new roles for women in the workplace. From 1910 to 1930 the percent of women working in agriculture decreased, while the percent of women in professional jobs increased, as shown in the table. $$ \begin{array}{|c|c|c|} \hline \text { Job type } & {\text { Percent holding that }} & {\text { Average percent increase }} \\ & {\text { job type in } 1910} & {\text { per year from } 1910 \text { to } 1930} \\ \hline \text { Agriculture } & {22.4 \%} & {-0.7 \%} \\ \hline \text { Professional } & {9.1 \%} & {0.25 \%} \\ \hline \end{array} $$ Assuming that both percentages can be represented by a linear function, use the information in the table above to estimate when the percent of women working in agriculture equaled the percent of women working in professional jobs between 1910 and 1930

Use the substitution method to solve the linear system. $$ \begin{array}{l} {-3 w+z=4} \\ {-9 w+5 z=-1} \end{array} $$

Plot the points and draw line segments connecting the points to create the polygon. Then write a system of linear inequalities that defines the polygonal region. Triangle: \((-2,0),(2,0),(0,2)\)

Use the substitution method or linear combinations to solve the linear system and tell how many solutions the system has. Then describe the graph of the system. $$\begin{aligned} &15 x-5 y=-20\\\ &-3 x+y=4 \end{aligned}$$

In Exercises 27–29, match the situation with the corresponding linear system. You have 7 packages of paper towels. Some packages have 3 rolls, but some have only 1 roll. There are 19 rolls altogether.

Use linear combinations to solve the linear system. Then check your solution. \(2 q=7-5 p\) \(4 p-16=q\)

You and your sister are saving money from your allowances. You have \(\$ 25\) and save \(\$ 3\) each week. Your sister has \(\$ 40\) and saves \(\$ 2\) each week. After how many weeks will you and your sister have the same amount of money?

Plot the points and draw line segments connecting the points to create the polygon. Then write a system of linear inequalities that defines the polygonal region. Rectangle: \((1,1),(7,1),(7,6),(1,6)\)

You are selling tickets for a high school play. Student tickets cost \(\$4\) and general admission tickets cost \(\$6\). You sell 525 tickets and collect \(\$2876\). Use the following verbal model to find how many of each type of ticket you sold.

Use the substitution method or linear combinations to solve the linear system and tell how many solutions the system has. Then describe the graph of the system. $$\begin{array}{l} {-6 x+2 y=-2} \\ {-4 x-y=8} \end{array}$$

Match the situation with the corresponding linear system. You buy 5 pairs of socks for 19 dollar. The wool socks cost 5 dollar per pair and the cotton socks cost 3 dollar per pair.

Use linear combinations to solve the linear system. Then check your solution. \(2 v=150-u\) \(2 u=150-v\)

You know how to solve the equation \(x+2=3 x-4\) algebraically. This equation can also be solved by graphing the following system of linear equations. $$ \begin{aligned} &y=x+2\\\ &y=3 x-4 \end{aligned} $$ a. Explain how the system of linear equations is related to the original equation given. b. Estimate the solution of the linear system graphically. c. Check that the \(x\) -coordinate from part (b) satisfies the original equation by substituting the \(x\) -coordinate for \(x\) in \(x+2=3 x-4\)

You are ordering softballs for two softball leagues. The size of a softball is measured by its circumference. The Pony League uses an 11 inch softball priced at \(\$3.50\). The Junior League uses a 12 inch softball priced at \(\$4.00\). The bill smeared in the rain, but you know the total was 80 softballs for \(\$305\). How many of each size did you order?

Plot the points and draw line segments connecting the points to create the polygon. Then write a system of linear inequalities that defines the polygonal region. Triangle: \((0,0),(-7,0),(-3,5)\)

Match the situation with the corresponding linear system. You have only 1 dollar bills and 5 dollar bills in your wallet. There are 7 bills worth a total of 19 dollar. $$ A. x+y=7 x+3 y=19$$ $$B. x+y=7 x+5 y=19$$ $$C. x+y=5 3 x+5 y=19$$

Use the substitution method or linear combinations to solve the linear system and tell how many solutions the system has. Then describe the graph of the system. $$\begin{array}{c} {2 x+y=-1} \\ {-6 x-3 y=-15} \end{array}$$

Use linear combinations to solve the linear system. Then check your solution. \(g-10 h=43\) \(18=-g+5 h\)

Which ordered pair is a solution of the following system of linear equations? $$ \begin{array}{r} {x+y=3} \\ {2 x+y=6} \end{array} $$ $$ \begin{array}{\underline{\phantom{xx}}}{ \text A {(0,3)}}\quad { \text B {(1,2)}}\quad \quad { \text C {(2,1)}}\quad \quad { \text D {(3,0)}} \end{array} $$

The rectangle at the right has a perimeter of 40 centimeters. The length of the rectangle is 4 times as long as the width. Find the dimensions of the rectangle.

Plot the points and draw line segments connecting the points to create the polygon. Then write a system of linear inequalities that defines the polygonal region. Trapezoid: \((-1,1),(1,3),(4,3),(6,1)\)

A contracting company rents a generator for 6 hours and a heavy-duty saw for 6 hours at a total cost of \( 48 . \)For another job the company rents the generator for 4 hours and the saw for 8 hours for a total cost of \(40 .\) Find the hourly rates \(g\) (for the generator) and \(s\) (for the saw) by solving the system of equations \(6 g+6 s=48\) and \(4 g+8 s=40\)

Use linear combinations to solve the linear system. Then check your solution. \(5 s+8 t=70\) \(60=5 s-8 t\)

Use the following information. You are planning the menu for your restaurant. For Saturday night you plan to serve roast beef and teriyaki chicken. You expect to serve at least 240 pounds of meat that evening and that less beef will be ordered than chicken. The roast beef costs 5 dollars per pound and the chicken costs 3 dollars per pound. You have a budget of at most 1200 dollars for meat for Saturday night. Copy and complete the following system of linear inequalities that shows the pounds \(b\) of roast beef meals and the pounds \(c\) of teriyaki chicken meals that you could prepare for Saturday night. $$ b+c \geq ? $$ $$ b ? c $$ $$ ? \cdot b+? \cdot c \leq 1200 $$

One share of ABC stock is worth three times as much as XYZ stock. An investor has 100 shares of each. If the total value of the stocks is \(\$4500\), how much money is invested in each stock?

You designate one row in your garden to broccoli and pea plants. Each broccoli plant needs 12 inches of space and each pea plant needs 6 inches of space. The row is 10 feet (120 inches) long. If you want a total of 13 plants, how many of each plant can you have?

You have a necklace and matching bracelet with 2 types of beads. There are 40 small beads and 6 large beads on the necklace. The bracelet has 20 small beads and 3 large beads. The necklace weighs 9.6 grams and the bracelet weighs 4.8 grams. If the threads holding the beads have no significant weight, can you find the weight of one large bead? Explain.

Use linear combinations to solve the linear system. Then check your solution. \(x+2 y=5\) \(5 x-y=3\)