# 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

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?
6.25
Ciid
6.25
Big
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.
Ciid
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?
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.
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?
Yes
Val
chapter 1 foundations 1.2 exercises variables and algebraic symbols
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.
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.
i need help how to do this is confusing
what kind of math is it?
Danteii
help me to understand
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
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.
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?
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?
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?   By Edward Biton  By Sam Luong    By 