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

 Page 3 / 5

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.

rectangular field solutions
What is this?
Donna
the proudact of 3x^3-5×^2+3 and 2x^2+5x-4 in z7[x]/ is
?
Choli
a rock is thrown directly upward with an initial velocity of 96feet per second from a cliff 190 feet above a beach. The hight of tha rock above the beach after t second is given by the equation h=_16t^2+96t+190
Usman
Stella bought a dinette set on sale for $725. The original price was$1,299. To the nearest tenth of a percent, what was the rate of discount?
44.19%
Scott
40.22%
Terence
44.2%
Orlando
I don't know
Donna
if you want the discounted price subtract $725 from$1299. then divide the answer by $1299. you get 0.4419... but as percent you get 44.19... but to the nearest tenth... round .19 to .2 and you get 44.2% Orlando you could also just divide$725/$1299 and then subtract it from 1. then you get the same answer. Orlando p mulripied-5 and add 30 to it Tausif Reply p mulripied-5 and add30 Tausif p mulripied-5 and addto30 Tausif Can you explain further Monica Reply p mulripied-5 and add to 30 Tausif How do you find divisible numbers without a calculator? Jacob Reply TAKE OFF THE LAST DIGIT AND MULTIPLY IT 9. SUBTRACT IT THE DIGITS YOU HAVE LEFT. IF THE ANSWER DIVIDES BY 13(OR IS ZERO), THEN YOUR ORIGINAL NUMBER WILL ALSO DIVIDE BY 13!IS DIVISIBLE BY 13 BAINAMA When she graduates college, Linda will owe$43,000 in student loans. The interest rate on the federal loans is 4.5% and the rate on the private bank loans is 2%. The total interest she owes for one year was $1,585. What is the amount of each loan? Ariana Reply Sean took the bus from Seattle to Boise, a distance of 506 miles. If the trip took 7 2/3 hours, what was the speed of the bus? Kirisma Reply 66miles/hour snigdha How did you work it out? Esther s=mi/hr 2/3~0.67 s=506mi/7.67hr = ~66 mi/hr Orlando hello, I have algebra phobia. Subtracting negative numbers always seem to get me confused. Alicia Reply what do you need help in? Felix subtracting a negative....is adding!! Heather look at the numbers if they have different signs, it's like subtracting....but you keep the sign of the largest number... Felix for example.... -19 + 7.... different signs...subtract.... 12 keep the sign of the "largest" number 19 is bigger than 7.... 19 has the negative sign... Therefore, -12 is your answer... Felix —12 Niazmohammad Thanks Felix.l also get confused with signs. Esther Thank you for this Shatey ty Graham think about it like you lost$19 (-19), then found $7(+7). Totally you lost just$12 (-12)
Annushka
I used to struggle a lot with negative numbers and math in general what I typically do is look at it in terms of money I have -$5 in my account I then take out 5 more dollars how much do I have in my account well-$10 ... I also for a long time would draw it out on a number line to visualize it
Meg
practicing with smaller numbers to understand then working with larger numbers helps too and the song/rhyme same sign add and keep opposite signs subtract keep the sign of the bigger # then you'll be exact
Meg
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
168.50=R
Heather
john is 5years older than wanjiru.the sum of their years is27years.what is the age of each
46
mustee
j 17 w 11
Joseph
john is 16. wanjiru is 11.
Felix
27-5=22 22÷2=11 11+5=16
Joyce
I don't see where the answers are.
Ed
Cindy and Richard leave their dorm in Charleston at the same time. Cindy rides her bicycle north at a speed of 18 miles per hour. Richard rides his bicycle south at a speed of 14 miles per hour. How long will it take them to be 96 miles apart?
3
Christopher
18t+14t=96 32t=96 32/96 3
Christopher
show that a^n-b^2n is divisible by a-b
What does 3 times your weight right now
Use algebra to combine 39×5 and the half sum of travel of 59+30
Cherokee
What is the segment of 13? Explain
Cherokee
my weight is 49. So 3 times is 147
Cherokee
kg to lbs you goin to convert 2.2 or one if the same unit your going to time your body weight by 3. example if my body weight is 210lb. what would be my weight if I was 3 times as much in kg. that's you do 210 x3 = 630lb. then 630 x 2.2= .... hope this helps
tyler
How to convert grams to pounds?
paul