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Solve the system by substitution. { 2 x 3 y = 12 −12 y + 8 x = 48

infinitely many solutions

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Solve the system by substitution. { 5 x + 2 y = 12 −4 y 10 x = −24

infinitely many solutions

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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. { 5 x 2 y = −10 y = 5 2 x

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 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.

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Solve the system by substitution. { 3 x + 2 y = 9 y = 3 2 x + 1

no solution

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Solve the system by substitution. { 5 x 3 y = 2 y = 5 3 x 4

no solution

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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 = 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 9 2 and 9 2 .
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The sum of two numbers is 10. One number is 4 less than the other. Find the numbers.

The numbers are 3 and 7.

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

The numbers are 2 and −8.

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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.
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Questions & Answers

Aziza is solving this equation-2(1+x)=4x+10
Sechabe Reply
No. 3^32 -1 has exactly two divisors greater than 75 and less than 85 what is their product?
KAJAL Reply
x^2+7x-19=0 has Two solutions A and B give your answer to 3 decimal places
Adedamola Reply
please the answer to the example exercise
Patricia Reply
3. When Jenna spent 10 minutes on the elliptical trainer and then did circuit training for20 minutes, her fitness app says she burned 278 calories. When she spent 20 minutes onthe elliptical trainer and 30 minutes circuit training she burned 473 calories. How manycalories does she burn for each minute on the elliptical trainer? How many calories doesshe burn for each minute of circuit training?
Edwin Reply
.473
Angelita
?
Angelita
John left his house in Irvine at 8:35 am to drive to a meeting in Los Angeles, 45 miles away. He arrived at the meeting at 9:50. At 3:30 pm, he left the meeting and drove home. He arrived home at 5:18.
DaYoungan Reply
p-2/3=5/6 how do I solve it with explanation pls
Adedamola Reply
P=3/2
Vanarith
1/2p2-2/3p=5p/6
James
don't understand answer
Cindy
4.5
Ruth
is y=7/5 a solution of 5y+3=10y-4
Adedamola Reply
yes
James
don't understand answer
Cindy
Lucinda has a pocketful of dimes and quarters with a value of $6.20. The number of dimes is 18 more than 3 times the number of quarters. How many dimes and how many quarters does Lucinda have?
Rhonda Reply
Find an equation for the line that passes through the point P ( 0 , − 4 ) and has a slope 8/9 .
Gabriel Reply
is that a negative 4 or positive 4?
Felix
y = mx + b
Felix
if negative -4, then -4=8/9(0) + b
Felix
-4=b
Felix
if positive 4, then 4=b
Felix
then plug in y=8/9x - 4 or y=8/9x+4
Felix
Macario is making 12 pounds of nut mixture with macadamia nuts and almonds. macadamia nuts cost $9 per pound and almonds cost $5.25 per pound. how many pounds of macadamia nuts and how many pounds of almonds should macario use for the mixture to cost $6.50 per pound to make?
Cherry Reply
Nga and Lauren bought a chest at a flea market for $50. They re-finished it and then added a 350 % mark - up
Makaila Reply
$1750
Cindy
the sum of two Numbers is 19 and their difference is 15
Abdulai Reply
2, 17
Jose
interesting
saw
4,2
Cindy
Felecia left her home to visit her daughter, driving 45mph. Her husband waited for the dog sitter to arrive and left home 20 minutes, or 13 hour later. He drove 55mph to catch up to Felecia. How long before he reaches her?
Rafi Reply
hola saben como aser un valor de la expresión
NAILEA
integer greater than 2 and less than 12
Emily Reply
2 < x < 12
Felix
I'm guessing you are doing inequalities...
Felix
Actually, translating words into algebraic expressions / equations...
Felix
hi
Darianna
hello
Mister
Eric here
Eric
6
Cindy

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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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