# 5.3 Solve systems of equations by elimination  (Page 2/6)

 Page 2 / 6

Once we get an equation with just one variable, we solve it. Then we substitute that value into one of the original equations to solve for the remaining variable. And, as always, we check our answer to make sure it is a solution to both of the original equations.

Now we’ll see how to use elimination to solve the same system of equations we solved by graphing and by substitution.

## How to solve a system of equations by elimination

Solve the system by elimination. $\left\{\begin{array}{c}2x+y=7\hfill \\ x-2y=6\hfill \end{array}$

## Solution       Solve the system by elimination. $\left\{\begin{array}{c}3x+y=5\hfill \\ 2x-3y=7\hfill \end{array}$

$\left(2,-1\right)$

Solve the system by elimination. $\left\{\begin{array}{c}4x+y=-5\hfill \\ -2x-2y=-2\hfill \end{array}$

$\left(-2,3\right)$

The steps are listed below for easy reference.

## How to solve a system of equations by elimination.

1. Write both equations in standard form. If any coefficients are fractions, clear them.
2. Make the coefficients of one variable opposites.
• Decide which variable you will eliminate.
• Multiply one or both equations so that the coefficients of that variable are opposites.
3. Add the equations resulting from Step 2 to eliminate one variable.
4. Solve for the remaining variable.
5. Substitute the solution from Step 4 into one of the original equations. Then solve for the other variable.
6. Write the solution as an ordered pair.
7. Check that the ordered pair is a solution to both original equations.

First we’ll do an example where we can eliminate one variable right away.

Solve the system by elimination. $\left\{\begin{array}{c}x+y=10\hfill \\ x-y=12\hfill \end{array}$

## Solution Both equations are in standard form. The coefficients of y are already opposites. Add the two equations to eliminate y . The resulting equation has only 1 variable, x . Solve for x , the remaining variable. Substitute x = 11 into one of the original equations.  Solve for the other variable, y . Write the solution as an ordered pair. The ordered pair is (11, −1). Check that the ordered pair is a solution to both original equations. $\begin{array}{cccc}\begin{array}{ccc}\hfill x+y& =\hfill & 10\hfill \\ \hfill 11+\left(-1\right)& \stackrel{?}{=}\hfill & 10\hfill \\ \hfill 10& =\hfill & 10\phantom{\rule{0.2em}{0ex}}✓\hfill \end{array}& & & \begin{array}{ccc}\hfill x-y& =\hfill & 12\hfill \\ \hfill 11-\left(-1\right)& \stackrel{?}{=}\hfill & 12\hfill \\ \hfill 12& =\hfill & 12\phantom{\rule{0.2em}{0ex}}✓\hfill \end{array}\end{array}$ The solution is (11, −1).

Solve the system by elimination. $\left\{\begin{array}{c}2x+y=5\hfill \\ x-y=4\hfill \end{array}$

$\left(3,-1\right)$

Solve the system by elimination. $\left\{\begin{array}{c}x+y=3\hfill \\ -2x-y=-1\hfill \end{array}$

$\left(-2,5\right)$

In [link] , we will be able to make the coefficients of one variable opposites by multiplying one equation by a constant.

Solve the system by elimination. $\left\{\begin{array}{c}3x-2y=-2\hfill \\ 5x-6y=10\hfill \end{array}$

## Solution Both equations are in standard form. None of the coefficients are opposites. We can make the coefficients of y opposites by multiplying the first equation by −3. Simplify. Add the two equations to eliminate y . Solve for the remaining variable, x . Substitute x = −4 into one of the original equations.  Solve for y .   Write the solution as an ordered pair. The ordered pair is (−4, −5). Check that the ordered pair is a solution to both original equations. $\begin{array}{cccc}\begin{array}{ccc}\hfill 3x-2y& =\hfill & -2\hfill \\ \hfill 3\left(-4\right)-2\left(-5\right)& \stackrel{?}{=}\hfill & -2\hfill \\ \hfill -12+10& \stackrel{?}{=}\hfill & -2\hfill \\ \hfill -2y& =\hfill & -2\phantom{\rule{0.2em}{0ex}}✓\hfill \end{array}& & & \begin{array}{ccc}\hfill 5x-6y& =\hfill & 10\hfill \\ \hfill 3\left(-4\right)-6\left(-5\right)& \stackrel{?}{=}\hfill & 10\hfill \\ \hfill -20+30& \stackrel{?}{=}\hfill & 10\hfill \\ \hfill 10& =\hfill & 10\phantom{\rule{0.2em}{0ex}}✓\hfill \end{array}\end{array}$ The solution is (−4, −5).

Solve the system by elimination. $\left\{\begin{array}{c}4x-3y=1\hfill \\ 5x-9y=-4\hfill \end{array}$

$\left(1,1\right)$

Solve the system by elimination. $\left\{\begin{array}{c}3x+2y=2\hfill \\ 6x+5y=8\hfill \end{array}$

$\left(-2,4\right)$

Now we’ll do an example where we need to multiply both equations by constants in order to make the coefficients of one variable opposites.

Solve the system by elimination. $\left\{\begin{array}{c}4x-3y=9\hfill \\ 7x+2y=-6\hfill \end{array}$

## Solution

In this example, we cannot multiply just one equation by any constant to get opposite coefficients. So we will strategically multiply both equations by a constant to get the opposites. Both equations are in standard form. To get opposite coefficients of y , we will multiply the first equation by 2 and the second equation by 3. Simplify. Add the two equations to eliminate y . Solve for x . Substitute x = 0 into one of the original equations.  Solve for y .  Write the solution as an ordered pair. The ordered pair is (0, −3). Check that the ordered pair is a solution to both original equations. $\begin{array}{cccc}\begin{array}{ccc}\hfill 4x-3y& =\hfill & 9\hfill \\ \hfill 4\left(0\right)-3\left(-3\right)& \stackrel{?}{=}\hfill & 9\hfill \\ \hfill 9& =\hfill & 9\phantom{\rule{0.2em}{0ex}}✓\hfill \end{array}& & & \begin{array}{ccc}\hfill 7x+2y& =\hfill & -6\hfill \\ \hfill 7\left(0\right)+2\left(-3\right)& \stackrel{?}{=}\hfill & -6\hfill \\ \hfill -6& =\hfill & -6\phantom{\rule{0.2em}{0ex}}✓\hfill \end{array}\end{array}$ The solution is (0, −3).

What other constants could we have chosen to eliminate one of the variables? Would the solution be the same?

What is the lcm of 340
How many numbers each equal to y must be taken to make 15xy
15x
Martin
15x
Asamoah
15x
Hugo
1y
Tom
1y x 15y
Tom
find the equation whose roots are 1 and 2
(x - 2)(x -1)=0 so equation is x^2-x+2=0
Ranu
I believe it's x^2-3x+2
NerdNamedGerg
because the X's multiply by the -2 and the -1 and than combine like terms
NerdNamedGerg
find the equation whose roots are -1 and 4
Ans = ×^2-3×+2
Gee
find the equation whose roots are -2 and -1
(×+1)(×-4) = x^2-3×-4
Gee
there's a chatting option in the app wow
Nana
That's cool cool
Nana
Nice to meet you all
Nana
you too.
Joan
😃
Nana
Hey you all there are several Free Apps that can really help you to better solve type Equations.
Debra
Debra, which apps specifically. ..?
Nana
am having a course in elementary algebra ,any recommendations ?
samuel
Samuel Addai, me too at ucc elementary algebra as part of my core subjects in science
Nana
me too as part of my core subjects in R M E
Ken
at ABETIFI COLLEGE OF EDUCATION
Ken
ok great. Good to know.
Joan
5x + 1/3= 2x + 1/2
sanam
Plz solve this
sanam
5x - 3x = 1/2 - 1/3 2x = 1/6 x = 1/12
Ranu
Thks ranu
sanam
a trader gains 20 rupees loses 42 rupees and then gains ten rupees Express algebraically the result of his transactions
a trader gains 20 rupees loses 42 rupees and then gains 10 rupees Express algebraically the result of his three transactions
vinaya
a trader gains 20 rupees loses 42 rupees and then gains 10 rupees Express algebraically the result of his three transactions
vinaya
a trader gains 20 rupees loses 42 rupees and then gains 10 rupees Express algebraically the result of his three transactions
vinaya
Kim is making eight gallons of punch from fruit juice and soda. The fruit juice costs $6.04 per gallon and the soda costs$4.28 per gallon. How much fruit juice and how much soda should she use so that the punch costs $5.71 per gallon? Mohamed Reply (a+b)(p+q+r)(b+c)(p+q+r)(c+a) (p+q+r) muhammad Reply 4x-7y=8 2x-7y=1 what is the answer? Ramil Reply x=7/2 & y=6/7 Pbp x=7/2 & y=6/7 use Elimination Debra true bismark factoriz e usman 4x-7y=8 X=7/4y+2 and 2x-7y=1 x=7/2y+1/2 Peggie Ok cool answer peggie Frank thanks Ramil copy and complete the table. x. 5. 8. 12. then 9x-5. to the 2nd power+4. then 2xto the second power +3x Sandra Reply What is c+4=8 Penny Reply 2 Letha 4 Lolita 4 Rich 4 thinking C+4=8 -4 -4 C =4 thinking I need to study Letha 4+4=8 William During two years in college, a student earned$9,500. The second year, she earned $500 more than twice the amount she earned the first year. Nicole Reply 9500=500+2x Debra 9500-500=9000 9000÷2×=4500 X=4500 Debra X + Y = 9500....... & Y = 500 + 2X so.... X + 500 + 2X = 9500, them X = 3000 & Y = 6500 Pbp Bruce drives his car for his job. The equation R=0.575m+42 models the relation between the amount in dollars, R, that he is reimbursed and the number of miles, m, he drives in one day. Find the amount Bruce is reimbursed on a day when he drives 220 miles. Josh Reply Reiko needs to mail her Christmas cards and packages and wants to keep her mailing costs to no more than$500. The number of cards is at least 4 more than twice the number of packages. The cost of mailing a card (with pictures enclosed) is $3 and for a package the cost is$7.
hey
Juan
Sup
patrick
The sum of two numbers is 155. The difference is 23. Find the numbers
The sum of two numbers is 155. Their difference is 23. Find the numbers
Michelle
The difference between 89 and 66 is 23
Ciid
Joy is preparing 20 liters of a 25% saline solution. She only has 40% and 10% solution in her lab. How many liters of the 40% and how many liters of the 10% should she mix to make the 25% solution?
hello
bismark
I need a math tutor BAD
Stacie
Me too
Letha
me too
Xavier
ok
Bishal
teet
Bishal   By   By Rhodes By Dravida Mahadeo-J... By By 