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Section 3

Solving Systems of Equations by Elimination (Addition)

Fill in the blanks.The coefficients of $3 x$ and $-3 x$ are _______.

Fill in the blanks.When the given equations are added, the variable $y$ will be_______.$$\begin{array}{r}{5 x-6 y=10} \\{-3 x+6 y=24} \\\hline\end{array}$$

In the given system, which terms have coefficients that are opposites?$$\left\{\begin{array}{l}{3 x+7 y=-25} \\{4 x-7 y=12}\end{array}\right.$$

Fill in the blank. The objective of the elimination method is to obtain two equations whose sum will be one equation in one______.

Add each pair of equations..$$\begin{aligned} \text {a.} \quad&2 a+2 b=-6\\&3 a-2 b=2 \\\hline\end{aligned}$$$$\begin{array}{r}\text {b.} \quad{x-3 y=15} \\{-x-y=-14} \\\hline\end{array}$$

a. Multiply both sides of $4 x+y=2$ by 3.b. Multiply both sides of $x-3 y=4$ by $-2.$

If the elimination method is used to solve$$\left\{\begin{array}{l}{3 x+12 y=4} \\{6 x-4 y=7}\end{array}\right.$$a. By what would we multiply the first equation to eliminate the variable $x ?$b. By what would we multiply the second equation to eliminate the variable $y ?$

Suppose the following system is solved using the elimination method and it is found that $x$ is $2 .$ Find the value of $y .$$$\left\{\begin{array}{l}{4 x+3 y=11} \\{3 x-2 y=4}\end{array}\right.$$

What algebraic step should be performed toa. Clear $\frac{2}{3} x+4 y=-\frac{4}{5}$ of fractions?b. Clear $0.2 x-0.9 y=6.4$ of decimals?

a. Suppose $0=0$ is obtained when a system is solved by the elimination method. Does the system have a solution? Which of the following is a possible graph of the system?b. Suppose $0=2$ is obtained when a system is solved by the elimination method. Does the system have a solution? Which of the following is a possible graph of the system?

GRAPH CANNOT COPY.

Complete the solution to solve the system.$\text { Solve: }\left\{\begin{array}{l}{x+y=5} \\{x-y=-3}\end{array}\right.$$$\begin{aligned}&x+y=5\\&\begin{array}{r} {x-y=-3} \\\text{_______}{=2}\end{array}\end{aligned}$$$$x= \text{_______}$$$$\begin{array}{r}{x+y=5} \\ {+y=5} \\\text{_______}\end{array}$$$$y=\text{_______}$$

Write each equation in $A x+B y=C$ form:

$$\left\{\begin{array}{l}{7 x+y+3=0} \\{8 x+4=-y}\end{array}\right.$$

Use the elimination method to solve each system.$$\left\{\begin{array}{l}{x+y=5} \\{x-y=1}\end{array}\right.$$

Use the elimination method to solve each system.$$\left\{\begin{array}{l}{x-y=4} \\{x+y=8}\end{array}\right.$$

Use the elimination method to solve each system.$$\left\{\begin{array}{l}{x+y=-5} \\{-x+y=-1}\end{array}\right.$$

Use the elimination method to solve each system.$$\left\{\begin{array}{l}{-x+y=-3} \\{x+y=1}\end{array}\right.$$

Use the elimination method to solve each system.$$\left\{\begin{array}{l}{4 x+3 y=24} \\{4 x-3 y=-24}\end{array}\right.$$

Use the elimination method to solve each system.$$\left\{\begin{array}{l}{-9 x+5 y=-9} \\{-9 x-5 y=-9}\end{array}\right.$$

Use the elimination method to solve each system.$$\left\{\begin{array}{l}{2 s+t=-2} \\{-2 s-3 t=-6}\end{array}\right.$$

Use the elimination method to solve each system.$$\left\{\begin{array}{l}{-2 x+4 y=12} \\{2 x+4 y=28}\end{array}\right.$$

Use the elimination method to solve each system.$$\left\{\begin{array}{l}{x+3 y=-9} \\{x+8 y=-4}\end{array}\right.$$

Use the elimination method to solve each system.$$\left\{\begin{array}{l}{x+7 y=-22} \\{x+9 y=-24}\end{array}\right.$$

Use the elimination method to solve each system.$$\left\{\begin{array}{l}{7 x-y=10} \\{8 x-y=13}\end{array}\right.$$

Use the elimination method to solve each system.$$\left\{\begin{array}{l}{6 x-y=4} \\{9 x-y=10}\end{array}\right.$$

Use the elimination method to solve each system.$$\left\{\begin{array}{l}{7 x+4 y-14=0} \\{3 x=2 y-20}\end{array}\right.$$

Use the elimination method to solve each system.$$\left\{\begin{array}{l}{5 x-14 y-32=0} \\{-x=6 y+20}\end{array}\right.$$

Use the elimination method to solve each system.$$\left\{\begin{array}{l}{7 x-50 y+43=0} \\{x=4-3 y}\end{array}\right.$$

Use the elimination method to solve each system.$$\left\{\begin{array}{l}{x-2 y+1=0} \\{12 x=23-11 y}\end{array}\right.$$

Use the elimination method to solve each system.$$\left\{\begin{array}{l}{4 x+3 y=7} \\{3 x-2 y=-16}\end{array}\right.$$

Use the elimination method to solve each system.$$\left\{\begin{array}{l}{3 x-2 y=20} \\{2 x+7 y=5}\end{array}\right.$$

Use the elimination method to solve each system.$$\left\{\begin{array}{l}{5 a+8 b=2} \\{11 a-3 b=25}\end{array}\right.$$

Use the elimination method to solve each system.$$\left\{\begin{array}{l}{7 a-5 b=24} \\{12 a+8 b=8}\end{array}\right.$$

Use the elimination method to solve each system.$$\left\{\begin{array}{l}{\frac{1}{8} x-\frac{1}{8} y=\frac{3}{8}} \\{\frac{x}{4}+\frac{y}{4}=\frac{1}{2}}\end{array}\right.$$

Use the elimination method to solve each system.$$\left\{\begin{array}{l}{\frac{1}{8} x+\frac{1}{4} y=\frac{1}{4}} \\{\frac{x}{2}+\frac{y}{4}=\frac{1}{2}}\end{array}\right.$$

Use the elimination method to solve each system.$$\left\{\begin{array}{l}{\frac{3}{4} x-\frac{5}{8} y=\frac{1}{24}} \\{\frac{5 x}{6}-y=\frac{1}{4}}\end{array}\right.$$

Use the elimination method to solve each system.$$\left\{\begin{array}{l}{\frac{1}{3} x+\frac{1}{2} y=\frac{5}{3}} \\{\frac{x}{7}-\frac{y}{7}=-\frac{1}{7}}\end{array}\right.$$

Use the elimination method to solve each system. If there is no solution, or infinitely many solutions, so state.

$$\left\{\begin{array}{l}{3 x-5 y=-29} \\{3 x-5 y=15}\end{array}\right.$$

$$\left\{\begin{array}{l}{2 a-3 b=-6} \\{2 a-3 b=8}\end{array}\right.$$

$$\left\{\begin{array}{l}{3 x-16=5 y} \\{-3 x+5 y-33=0}\end{array}\right.$$

$$\left\{\begin{array}{l}{\frac{-18 x+y}{2}=\frac{7}{2}} \\{18 x=y}\end{array}\right.$$

$$\left\{\begin{array}{l}{0.4 x-0.7 y=-1.9} \\{-x+\frac{7 y}{4}=\frac{19}{4}}\end{array}\right.$$

$$\left\{\begin{array}{l}{0.1 x+2 y+0.2=0} \\{-\frac{x}{4}-5 y=\frac{1}{2}}\end{array}\right.$$

$$\left\{\begin{array}{l}{\frac{x-6 y}{2}=7} \\{-x+6 y+14=0}\end{array}\right.$$

$$\left\{\begin{array}{l}{2 x+5 y-13=0} \\{-2 x+13=5 y}\end{array}\right.$$

Solve the system by either the substitution or the elimination method.$$\left\{\begin{array}{l}{y=-3 x+9} \\{y=x+1}\end{array}\right.$$

Solve the system by either the substitution or the elimination method.$$\left\{\begin{array}{l}{x=5 y-4} \\{x=9 y-8}\end{array}\right.$$

Solve the system by either the substitution or the elimination method.$$\left\{\begin{array}{l}{4 x+6 y=5} \\{8 x-9 y=3}\end{array}\right.$$

Solve the system by either the substitution or the elimination method.$$\left\{\begin{array}{l}{3 a+4 b=36} \\{6 a-2 b=-21}\end{array}\right.$$

Solve the system by either the substitution or the elimination method.$$\left\{\begin{array}{l}{6 x-3 y=-7} \\{y+9 x=6}\end{array}\right.$$

Solve the system by either the substitution or the elimination method.$$\left\{\begin{array}{l}{9 x+4 y=31} \\{y-5=6 x}\end{array}\right.$$

Solve the system by either the substitution or the elimination method.$$\left\{\begin{array}{l}{x+y=1} \\{x-y=5}\end{array}\right.$$

Solve the system by either the substitution or the elimination method.$$\left\{\begin{array}{l}{x-y=-5} \\{x+y=1}\end{array}\right.$$

Solve the system by either the substitution or the elimination method.$$\left\{\begin{array}{c}{4(x-2 y)=36} \\{3 x-6 y=27}\end{array}\right.$$

Solve the system by either the substitution or the elimination method.$$\left\{\begin{aligned}2(x+2 y) &=15 \\3 x=8 &-6 y\end{aligned}\right.$$

Solve the system by either the substitution or the elimination method.$$\left\{\begin{array}{l}{x=y} \\{0.1 x+0.2 y=1.0}\end{array}\right.$$

Solve the system by either the substitution or the elimination method.$$\left\{\begin{array}{l}{x=y} \\{0.4 x-0.8 y=-0.5}\end{array}\right.$$

Solve the system by either the substitution or the elimination method.$$\left\{\begin{array}{l}{2 x+11 y=-10} \\{5 x+4 y=22}\end{array}\right.$$

$$\left\{\begin{array}{l}{3 x+4 y=12} \\{4 x+5 y=17}\end{array}\right.$$

Solve the system by either the substitution or the elimination method.$$\left\{\begin{array}{l}{7 x-21-6 y} \\{4 x+5 y=12}\end{array}\right.$$

Solve the system by either the substitution or the elimination method.$$\left\{\begin{array}{l}{-4 x=-3 y-13} \\{-6 x+8 y=-16}\end{array}\right.$$

Solve the system by either the substitution or the elimination method.$$\left\{\begin{array}{l}{9 x-10 y=0} \\{\frac{9 x-3 y}{63}=1}\end{array}\right.$$

Solve the system by either the substitution or the elimination method.$$\left\{\begin{array}{l}{8 x-9 y=0} \\{\frac{2 x-3 y}{6}=-1}\end{array}\right.$$

Solve the system by either the substitution or the elimination method.$$\left\{\begin{array}{l}{\frac{m}{4}+\frac{n}{3}=-\frac{1}{12}} \\{\frac{m}{2}-\frac{5}{4} n=\frac{7}{4}}\end{array}\right.$$

Solve the system by either the substitution or the elimination method.$$\left\{\begin{array}{l}{\frac{x}{2}-\frac{y}{3}=-2} \\{\frac{x}{3}+\frac{2}{3} y=\frac{4}{3}}\end{array}\right.$$

Solve the system by either the substitution or the elimination method.$$\left\{\begin{array}{l}{x-\frac{4}{3} y=\frac{1}{3}} \\{2 x+\frac{3}{2} y=\frac{1}{2}}\end{array}\right.$$

Solve the system by either the substitution or the elimination method.$$\left\{\begin{array}{l}{x+y=-\frac{1}{4}} \\{x-\frac{y}{2}=-\frac{3}{2}}\end{array}\right.$$

Solve the system by either the substitution or the elimination method.$$\left\{\begin{array}{l}{4 x-7 y+32=0} \\{5 x=4 y-2}\end{array}\right.$$

Solve the system by either the substitution or the elimination method.$$\left\{\begin{array}{l}{6 x=-3 y} \\{5 x+15=5 y}\end{array}\right.$$

$$\left\{\begin{array}{l}{3(x+4 y)=-12} \\{x=3 y+10}\end{array}\right.$$

Solve the system by either the substitution or the elimination method.$$\left\{\begin{array}{l}{3 x+2 y=3} \\{y=2(x-8)}\end{array}\right.$$

Solve the system by either the substitution or the elimination method.$$\left\{\begin{array}{l}{4 a+7 b=2} \\{9 a-3 b=1}\end{array}\right.$$

Solve the system by either the substitution or the elimination method.$$\left\{\begin{array}{l}{5 a-7 b=6} \\{7 a-6 b=8}\end{array}\right.$$

Solve the system by either the substitution or the elimination method.$$\left\{\begin{array}{l}{3 a-b=12.3} \\{4 a-b=14.9}\end{array}\right.$$

Solve the system by either the substitution or the elimination method.$$\left\{\begin{array}{l}{-7 x-y=8.5} \\{4 x-y=-12.4}\end{array}\right.$$

Solve the system by either the substitution or the elimination method.$$\left\{\begin{array}{l}{5 x-4 y=8} \\{-5 x-4 y=8}\end{array}\right.$$

Solve the system by either the substitution or the elimination method.$$\left\{\begin{array}{l}{2 r+s=-8} \\{-2 r+4 s=28}\end{array}\right.$$

Solve the system by either the substitution or the elimination method.$$\left\{\begin{array}{l}{9 a+16 b=-36} \\{7 a+4 b=48}\end{array}\right.$$

Solve the system by either the substitution or the elimination method.$$\left\{\begin{array}{l}{4 a+7 b=-24} \\{9 a+b=64}\end{array}\right.$$

Solve the system by either the substitution or the elimination method.$$\left\{\begin{array}{l}{8 x+12 y=-22} \\{3 x-2 y=8}\end{array}\right.$$

Solve the system by either the substitution or the elimination method.$$\left\{\begin{array}{l}{3 x+2 y=45} \\{5 x-4 y=20}\end{array}\right.$$

Solve the system by either the substitution or the elimination method.$$\left\{\begin{array}{l}{6 x+5 y+29=0} \\{0.02 x=0.03 y-0.05}\end{array}\right.$$

Solve the system by either the substitution or the elimination method.$$\left\{\begin{array}{l}{3 x=20 y+1} \\{0.04 x+0.05 y-0.33=0}\end{array}\right.$$

Solve the system by either the substitution or the elimination method.$$\left\{\begin{array}{l}{c=d-9} \\{5 c=3 d-35}\end{array}\right.$$

Solve the system by either the substitution or the elimination method.$$\left\{\begin{array}{l}{a=b+7} \\{3 a-15=5 b}\end{array}\right.$$

Solve the system by either the substitution or the elimination method.$$\left\{\begin{array}{l}{0.9 x+2.1=0.3 y} \\{0.4 x=0.7 y+1.9}\end{array}\right.$$

Solve the system by either the substitution or the elimination method.$$\left\{\begin{array}{l}{0.7 x+1.1=0.4 y} \\{0.4 x=0.7 y+2.2}\end{array}\right.$$

Solve the system by either the substitution or the elimination method.$$\left\{\begin{array}{l}{5 c+2 d=-5} \\{6 c+2 d=-10}\end{array}\right.$$

Solve the system by either the substitution or the elimination method.$$\left\{\begin{array}{l}{11 c+3 d=-68} \\{10 c+3 d=-64}\end{array}\right.$$

Solve the system by either the substitution or the elimination method.$$\left\{\begin{array}{l}{\frac{2}{15} x-\frac{1}{5} y=\frac{1}{3}} \\{\frac{2}{15} x-\frac{1}{5} y=\frac{1}{10}}\end{array}\right.$$

Solve the system by either the substitution or the elimination method.$$\left\{\begin{array}{l}{\frac{1}{5} x+\frac{3}{5} y=\frac{4}{5}} \\{\frac{1}{6} x+\frac{1}{2} y=\frac{2}{3}}\end{array}\right.$$

Education. The graph shows educational trends during the years $1980-2009$ for persons 25 years or older in the United States. The equation $9 x+11 y=352$ approximates the percent $y$ that had less than high school completion. The equation $5 x-11 y=-198$ approximates the percent $y$ that had a Bachelor's or higher degree. In each casc, $x$ is the number of years since $1980 .$ Use the elimination method to determine in what year the percents were equal.

Newspapers. The graph shows the trends in the newspaper publishing industry during the years $1990-2008$ in the United States. The equation $37 x-2 y=-1,128$ models the number $y$ of morning newspapers published and 3 $1 x+y=1,059$ models the number $y$ of evening newspapers published. In each case, $x$ is the number of years since $1990 .$ Use the elimination method to determine in what year there were an equal number of morning and evening newspapers being published.

CFL Bulbs. The graph below shows how a more expensive, but more energy-efficient, compact fluorescent light bulb eventually costs less to use than an incandescent light bulb. The equation $60 c-d=96$ approximates the cost $c$ (in dollars) to purchase and use a CFL. bulb 8 hours a day for $d$ days. The equation $15 c-d=6$ does the same for an incandescent bulb. Use the elimination method to determine after how many days the upgrade to a CFL bulb begins to save money.

The Human Skeleton. The equation $h+f=53$ models the fact that the number of bones in the hand and foot totals $53 .$ The equation $h-f=1$ models the fact that the difference between the number of bones in the hand and foot is just $1 .$ Use the elimination method to find $h$ and $f.$

Why is the method for solving systems that is discussed in this section called the elimination method? Why is it also referred to as the addition method?

If the elimination method is to be used to solve this system, what is wrong with the form in which it is written?$$\left\{\begin{array}{c}{2 x-5 y=-3} \\{-2 y-10=-5 x}\end{array}\right.$$

Can the system $\left\{\begin{array}{l}{2 x+3 y=13} \\ {7 x-3 y=-5}\end{array}\right.$ be solved more easily using \right. the elimination method or the substitution method? Explain.

Explain the error in the following workSolve:$$\left\{\begin{array}{rlrl}{x+y} & {=1} & {x+y} & {=1} \\ {x-y} & {=5} & {\frac{+x-y}{2 x}} & {=6}\end{array}\right.\frac{2 x}{2}=\frac{6}{2}$$

$$x=3$$

The solution is 3

Find an equation of the line with slope $-\frac{11}{6}$ that passes through $(2,-6) .$ Write the equation in slope-intercept form.

Solve $S=2 \pi r h+2 \pi r^{2}$ for $h$

Evaluate: $-10\left(18-4^{2}\right)^{3}$

Evaluate: $-5^{2}$

Use the elimination method to solve each system.$$\left\{\begin{array}{l}{\frac{x-3}{2}=\frac{11}{6}-\frac{y+5}{3}} \\{\frac{x+3}{3}-\frac{y+3}{4}=\frac{5}{12}}\end{array}\right.$$

Use the elimination method to solve each system.$$\left\{\begin{array}{l}{\frac{4(x+1)}{34}-\frac{1}{2}-\frac{3(y-1)}{34}} \\{0.2(x+0.2)+0.3(y-0.3)=0.75}\end{array}\right.$$