# 5.2 Solve systems of equations by substitution  (Page 3/5)

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Solve the system by substitution. $\left\{\begin{array}{c}2x-3y=12\hfill \\ -12y+8x=48\hfill \end{array}$

infinitely many solutions

Solve the system by substitution. $\left\{\begin{array}{c}5x+2y=12\hfill \\ -4y-10x=-24\hfill \end{array}$

infinitely many solutions

Look back at the equations in [link] . Is there any way to recognize that they are the same line?

Let’s see what happens in the next example.

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

## Solution

The second equation is already solved for y , so we can substitute for y in the first equation.

 Substitute x for y in the first equation. Replace the y with $\frac{5}{2}x.$ Solve for x .

Since 0 = −10 is a false statement the equations are inconsistent. The graphs of the two equation would be parallel lines. The system has no solutions.

Solve the system by substitution. $\left\{\begin{array}{c}3x+2y=9\hfill \\ y=-\frac{3}{2}x+1\hfill \end{array}$

no solution

Solve the system by substitution. $\left\{\begin{array}{c}5x-3y=2\hfill \\ y=\frac{5}{3}x-4\hfill \end{array}$

no solution

## Solve applications of systems of equations by substitution

We’ll copy here the problem solving strategy we used in the Solving Systems of Equations by Graphing section for solving systems of equations. Now that we know how to solve systems by substitution, that’s what we’ll do in Step 5.

## How to use a problem solving strategy for systems of linear equations.

1. Read the problem. Make sure all the words and ideas are understood.
2. Identify what we are looking for.
3. Name what we are looking for. Choose variables to represent those quantities.
4. Translate into a system of equations.
5. Solve the system of equations using good algebra techniques.
6. Check the answer in the problem and make sure it makes sense.
7. Answer the question with a complete sentence.

Some people find setting up word problems with two variables easier than setting them up with just one variable. Choosing the variable names is easier when all you need to do is write down two letters. Think about this in the next example—how would you have done it with just one variable?

The sum of two numbers is zero. One number is nine less than the other. Find the numbers.

## Solution

 Step 1. Read the problem. Step 2. Identify what we are looking for. We are looking for two numbers. Step 3. Name what we are looking for. Let $n=$ the first number Let $m=$ the second number Step 4. Translate into a system of equations. The sum of two numbers is zero. One number is nine less than the other. The system is: Step 5. Solve the system of equations. We will use substitution since the second equation is solved for n . Substitute m − 9 for n in the first equation. Solve for m . Substitute $m=\frac{9}{2}$ into the second equation and then solve for n . Step 6. Check the answer in the problem. Do these numbers make sense in the problem? We will leave this to you! Step 7. Answer the question. The numbers are $\frac{9}{2}$ and $-\frac{9}{2}.$

The sum of two numbers is 10. One number is 4 less than the other. Find the numbers.

The numbers are 3 and 7.

The sum of two number is −6. One number is 10 less than the other. Find the numbers.

The numbers are 2 and −8.

In the [link] , we’ll use the formula for the perimeter of a rectangle, P = 2 L + 2 W .

The perimeter of a rectangle is 88. The length is five more than twice the width. Find the length and the width.

## Solution

 Step 1. Read the problem. Step 2. Identify what you are looking for. We are looking for the length and width. Step 3. Name what we are looking for. Let $L=$ the length    $W=$ the width Step 4. Translate into a system of equations. The perimeter of a rectangle is 88. 2 L + 2 W = P The length is five more than twice the width. The system is: Step 5. Solve the system of equations. We will use substitution since the second equation is solved for L . Substitute 2 W + 5 for L in the first equation. Solve for W . Substitute W = 13 into the second equation and then solve for L . Step 6. Check the answer in the problem. Does a rectangle with length 31 and width 13 have perimeter 88? Yes. Step 7. Answer the equation. The length is 31 and the width is 13.

#### Questions & Answers

4x+7y=29,x+3y=11 substitute method of linear equation
Srinu Reply
substitute method of linear equation
Srinu
Solve one equation for one variable. Using the 2nd equation, x=11-3y. Substitute that for x in first equation. this will find y. then use the value for y to find the value for x.
bruce
I want to learn
Elizebeth
help
Elizebeth
I want to learn. Please teach me?
Wayne
1) Use any equation, and solve for any of the variables. Since the coefficient of x (the number in front of the x) in the second equation is 1 (it actually isn't shown, but 1 * x = x), use that equation. Subtract 3y from both sides (this isolates the x on the left side of the equal sign).
bruce
2) This results in x=11-3y. x is note in terms of y. Use that as the value of x and substitute for all x in the first equation. The first equation becomes 4(11-3y)+7y =29. Note that the only variable left in the first equation is the y. If you have multiple variable, then something is wrong.
bruce
3) Distribute (multiply) the 4 across 11-3y to get 44-12y. Add this to the 7y. So, the equation is now 44-5y=29.
bruce
4) Solve 44-5y=29 for y. Isolate the y by subtracting 44 from birth sides, resulting in -5y=-15. Now, divide birth sides by -5 (since you have -5y). This results in y=3. You now have the value of one variable.
bruce
5) The last step is to take the value of y from Step 4) and substitute into the 2nd equation. Therefore: x+3y=11 becomes x+3(3)=11. Then multiplying, x+9=11. Finally, solve for x by subtracting 9 from both sides. Therefore, x=2.
bruce
6) The ordered pair of (2, 3) is the proposed solution. To check, substitute those values into either equation. If the result is true, then the solution is correct. 4(2)+7(3)=8+21=29. TRUE! Finished.
bruce
At 1:30 Marlon left his house to go to the beach, a distance of 5.625 miles. He rose his skateboard until 2:15, and then walked the rest of the way. He arrived at the beach at 3:00. Marlon's speed on his skateboard is 1.5 times his walking speed. Find his speed when skateboarding and when walking.
Andrew Reply
divide 3x⁴-4x³-3x-1 by x-3
Ritik Reply
how to multiply the monomial
Ceny Reply
Two sisters like to compete on their bike rides. Tamara can go 4 mph faster than her sister, Samantha. If it takes Samantha 1 hours longer than Tamara to go 80 miles, how fast can Samantha ride her bike? Got questions? Get instant answers now!
Seera Reply
how do u solve that question
Seera
Two sisters like to compete on their bike rides. Tamara can go 4 mph faster than her sister, Samantha. If it takes Samantha 1 hours longer than Tamara to go 80 miles, how fast can Samantha ride her bike?
Seera
Speed=distance ÷ time
Tremayne
x-3y =1; 3x-2y+4=0 graph
Juned Reply
Brandon has a cup of quarters and dimes with a total of 5.55\$. The number of quarters is five less than three times the number of dimes
ashley Reply
app is wrong how can 350 be divisible by 3.
Raheem Reply
June needs 48 gallons of punch for a party and has two different coolers to carry it in. The bigger cooler is five times as large as the smaller cooler. How many gallons can each cooler hold?
Susanna Reply
Susanna if the first cooler holds five times the gallons then the other cooler. The big cooler holda 40 gallons and the 2nd will hold 8 gallons is that correct?
Georgie
@Susanna that person is correct if you divide 40 by 8 you can see it's 5 it's simple
Ashley
@Geogie my bad that was meant for u
Ashley
Hi everyone, I'm glad to be connected with you all. from France.
Lorris Reply
I'm getting "math processing error" on math problems. Anyone know why?
Ray Reply
Can you all help me I don't get any of this
Jade Reply
4^×=9
Alberto Reply
Did anyone else have trouble getting in quiz link for linear inequalities?
Sireka Reply
operation of trinomial
Justin Reply

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