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By the end of this section, you will be able to:
  • Solve a system of equations by substitution
  • Solve applications of systems of equations by substitution

Before you get started, take this readiness quiz.

  1. Simplify −5 ( 3 x ) .
    If you missed this problem, review [link] .
  2. Simplify 4 2 ( n + 5 ) .
    If you missed this problem, review [link] .
  3. Solve for y . 8 y 8 = 32 2 y
    If you missed this problem, review [link] .
  4. Solve for x . 3 x 9 y = −3
    If you missed this problem, review [link] .

Solving systems of linear equations by graphing is a good way to visualize the types of solutions that may result. However, there are many cases where solving a system by graphing is inconvenient or imprecise. If the graphs extend beyond the small grid with x and y both between −10 and 10, graphing the lines may be cumbersome. And if the solutions to the system are not integers, it can be hard to read their values precisely from a graph.

In this section, we will solve systems of linear equations by the substitution method.

Solve a system of equations by substitution

We will use the same system we used first for graphing.

{ 2 x + y = 7 x 2 y = 6

We will first solve one of the equations for either x or y . We can choose either equation and solve for either variable—but we’ll try to make a choice that will keep the work easy.

Then we substitute that expression into the other equation. The result is an equation with just one variable—and we know how to solve those!

After we find the value of one variable, we will substitute that value into one of the original equations and solve for the other variable. Finally, we check our solution and make sure it makes both equations true.

We’ll fill in all these steps now in [link] .

How to solve a system of equations by substitution

Solve the system by substitution. { 2 x + y = 7 x 2 y = 6

 

 

 

Solution

This figure has three columns and six rows. The first row says, “Step 1. Solve one of the equations for either variable.” To the right of this, the middl row reads, “We’ll solve the first equation for y.” The third column shows the two equations: 2x + y = 7 and x – 2y = 6. It shows that 2x + y = 7 becomes y = 7 – 2x. The second row reads, “Step 2. Substitute the expression from Step 1 into the other equation.” Then, “We replace y in the second equation with the expression 7 – 2x.” It then shows the x – 2y = 6 becomes x – 2(7 – 2x) = 6. The third row says, “Step 3: Solve the resulting equation.” Then “Now we have an equation with just 1 variable. WE know how to solve this!” It then shows that x – 2(7 – 2x) = 6 becomes x – 14 + 4x = 6 which becomes 5x = 20. Thus x = 4. The fourth row says, “Step 4. Substitute the solution in Step 3 into one of the original quaitons to find the other variable.” Then, “We’ll use the first equation and replace x with 4.” Then it shows that 2x + y = 7 becomes 2(4) + y = 7. This becomes 8 + y = 7, and thus y = −1. The fifth row reads, “Step 5. Write the solution as an ordered pair.” Then “The ordered air is (x, y).” Then (4, −1). The sixth row reads, “Step 6. Check that the order pair is a solution to both original equations.” Then, “Substitute (4, −1) into both equations and make sure they are both true.” It then shows that 2x + y = 7 becomxe 2(4) + −1 = 7, and thus 7 = 7. It also shows that x – 2y = 6 becomes 4 – 2(−1) = 6, and thus 6−6. It also states, “Both equations are ture. (4, −1) is the solution to the system.”
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Solve the system by substitution. { −2 x + y = −11 x + 3 y = 9

( 6 , 1 )

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Solve the system by substitution. { x + 3 y = 10 4 x + y = 18

( 4 , 2 )

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Solve a system of equations by substitution.

  1. Solve one of the equations for either variable.
  2. Substitute the expression from Step 1 into the other equation.
  3. Solve the resulting equation.
  4. Substitute the solution in Step 3 into one of the original equations to find the other variable.
  5. Write the solution as an ordered pair.
  6. Check that the ordered pair is a solution to both original equations.

If one of the equations in the system is given in slope–intercept form, Step 1 is already done! We’ll see this in [link] .

Solve the system by substitution.

{ x + y = −1 y = x + 5

Solution

The second equation is already solved for y . We will substitute the expression in place of y in the first equation.

.
The second equation is already solved for y .
We will substitute into the first equation.
Replace the y with x + 5. .
Solve the resulting equation for x . .
.
.
Substitute x = −3 into y = x + 5 to find y . .
.
The ordered pair is (−3, 2). .
Check the ordered pair in both equations:

x + y = 1 3 + 2 = ? 1 1 = 1 y = x + 5 2 = ? 3 + 5 2 = 2
The solution is (−3, 2).

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Solve the system by substitution. { x + y = 6 y = 3 x 2

( 2 , 4 )

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Solve the system by substitution. { 2 x y = 1 y = −3 x 6

( −1 , −3 )

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If the equations are given in standard form, we’ll need to start by solving for one of the variables. In this next example, we’ll solve the first equation for y .

Questions & Answers

Larry and Tom were standing next to each other in the backyard when Tom challenged Larry to guess how tall he was. Larry knew his own height is 6.5 feet and when they measured their shadows, Larry’s shadow was 8 feet and Tom’s was 7.75 feet long. What is Tom’s height?
genevieve Reply
6.25
Ciid
Wayne is hanging a string of lights 57 feet long around the three sides of his patio, which is adjacent to his house. the length of his patio, the side along the house, is 5 feet longer than twice it's width. Find the length and width of the patio.
Katherine Reply
complexed. could you please help me figure out?
Ciid
(sin=opp/adj) (tan= opp/adj) cos=hyp/adj
tyler
(sin=opp/adj) (tan= opp/adj) cos=hyp/adj dont quote me on it look it up
tyler
(sin=opp/adj) (tan= opp/adj) cos=hyp/adj dont quote me on it look it up
tyler
(sin=opp/adj) (tan= opp/adj) cos=hyp/adj dont quote me on it look it up
tyler
SOH = Sine is Opposite over Hypotenuse. CAH= Cosine is Adjacent over Hypotenuse. TOA = Tangent is Opposite over Adjacent.
tyler
H=57 and O=285 figure out what the adjacent?
tyler
Amara currently sells televisions for company A at a salary of $17,000 plus a $100 commission for each television she sells. Company B offers her a position with a salary of $29,000 plus a $20 commission for each television she sells. How many televisions would Amara need to sell for the options to be equal?
Marisol Reply
what is the quantity and price of the televisions for both options?
karl
Amara has to sell 120 televisions to make 29,000 of the salary of company B. 120 * TV 100= commission 12,000+ 17,000 = of company a salary 29,000 17000+
Ciid
Amara has to sell 120 televisions to make 29,000 of the salary of company B. 120 * TV 100= commission 12,000+ 17,000 = of company a salary 29,000
Ciid
I'm mathematics teacher from highly recognized university.
Mzo Reply
here a question professor How many soldiers are there in a group of 27 sailors and soldiers if there are four fifths as many sailors as soldiers? can you write out the college you went to with the name of the school you teach at and let me know the answer I've got it to be honest with you
tyler
is anyone else having issues with the links not doing anything?
Helpful Reply
Yes
Val
chapter 1 foundations 1.2 exercises variables and algebraic symbols
theresa Reply
June needs 45 gallons of punch for a party and has 2 different coolers to carry it in. The bigger cooler is 5 times as large as the smaller cooler. How many gallons can each cooler hold? Enter the answers in decimal form.
Samer Reply
Joseph would like to make 12 pounds of a coffee blend at a cost of $6.25 per pound. He blends Ground Chicory at $4.40 a pound with Jamaican Blue Mountain at $8.84 per pound. How much of each type of coffee should he use?
Samer
4x6.25= $25 coffee blend 4×4.40= $17.60 ground chicory 4x8.84= 35.36 blue mountain. In total they will spend for 12 pounds $77.96 they will spend in total
tyler
DaMarcus and Fabian live 23 miles apart and play soccer at a park between their homes. DaMarcus rode his bike for three-quarters of an hour and Fabian rode his bike for half an hour to get to the park. Fabian’s speed was six miles per hour faster than DaMarcus’ speed. Find the speed of both soccer players.
Sage Reply
i need help how to do this is confusing
Alvina Reply
what kind of math is it?
Danteii
help me to understand
Alvina Reply
huh, what is the algebra problem
Daniel
How many soldiers are there in a group of 27 sailors and soldiers if there are four fifths many sailors as soldiers?
tyler
What is the domain and range of heaviside
Christopher Reply
What is the domain and range of Heaviside and signum
Christopher
25-35
Fazal
The hypotenuse of a right triangle is 10cm long. One of the triangle’s legs is three times the length of the other leg. Find the lengths of the three sides of the triangle.
Edi Reply
Tickets for a show are $70 for adults and $50 for children. For one evening performance, a total of 300 tickets were sold and the receipts totaled $17,200. How many adult tickets and how many child tickets were sold?
Mum Reply
A 50% antifreeze solution is to be mixed with a 90% antifreeze solution to get 200 liters of a 80% solution. How many liters of the 50% solution and how many liters of the 90% solution will be used?
Edi Reply
June needs 45 gallons of punch for a party and has 2 different coolers to carry it in. The bigger cooler is 5 times as large as the smaller cooler. How many gallons can each cooler hold?
Jesus Reply

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Source:  OpenStax, Elementary algebra. OpenStax CNX. Jan 18, 2017 Download for free at http://cnx.org/content/col12116/1.2
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