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Solve the system by substitution. { 3 x + y = 5 2 x + 4 y = −10

Solution

We need to solve one equation for one variable. Then we will substitute that expression into the other equation.

Solve for y .

Substitute into the other equation.
.
Replace the y with −3 x + 5. .
Solve the resulting equation for x . .
.
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Substitute x = 3 into 3 x + y = 5 to find y . .
.
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The ordered pair is (3, −4). .
Check the ordered pair in both equations:

3 x + y = 5 3 · 3 + ( 4 ) = ? 5 9 4 = ? 5 5 = 5 2 x + 4 y = 10 2 · 3 + 4 ( 4 ) = 10 6 16 = ? 10 10 = 10
The solution is (3, −4).

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Solve the system by substitution. { 4 x + y = 2 3 x + 2 y = −1

( 1 , −2 )

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

( 2 , 6 )

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In [link] it was easiest to solve for y in the first equation because it had a coefficient of 1. In [link] it will be easier to solve for x .

Solve the system by substitution. { x 2 y = −2 3 x + 2 y = 34

Solution

We will solve the first equation for x and then substitute the expression into the second equation.

.
Solve for x .

Substitute into the other equation.
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Replace the x with 2 y − 2. .
Solve the resulting equation for y . .

Substitute y = 5 into x − 2 y = −2 to find x .
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The ordered pair is (8, 5).
Check the ordered pair in both equations:

x 2 y = 2 8 2 · 5 = ? 2 8 10 = ? 2 2 = 2 3 x + 2 y = 34 3 · 8 + 2 · 5 = ? 34 24 + 10 = ? 34 34 = 34
The solution is (8, 5).

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Solve the system by substitution. { x 5 y = 13 4 x 3 y = 1

( −2 , −3 )

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Solve the system by substitution. { x 6 y = −6 2 x 4 y = 4

( 6 , 2 )

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When both equations are already solved for the same variable, it is easy to substitute!

Solve the system by substitution. { y = −2 x + 5 y = 1 2 x

Solution

Since both equations are solved for y , we can substitute one into the other.

Substitute 1 2 x for y in the first equation. .
Replace the y with 1 2 x . .
Solve the resulting equation. Start
by clearing the fraction.
.
Solve for x . .
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Substitute x = 2 into y = 1 2 x to find y . .
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The ordered pair is (2,1).
Check the ordered pair in both equations:

y = 1 2 x 1 = ? 1 2 · 2 1 = 1 y = 2 x + 5 1 = ? 2 · 2 + 5 1 = 4 + 5 1 = 1
The solution is (2,1).

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Solve the system by substitution. { y = 3 x 16 y = 1 3 x

( 6 , 2 )

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Solve the system by substitution. { y = x + 10 y = 1 4 x

( 8 , 2 )

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Be very careful with the signs in the next example.

Solve the system by substitution. { 4 x + 2 y = 4 6 x y = 8

Solution

We need to solve one equation for one variable. We will solve the first equation for y .

.
Solve the first equation for y . .
Substitute −2 x + 2 for y in the second equation. .
Replace the y with −2 x + 2. .
Solve the equation for x . .
.
.


Substitute x = 5 4 into 4 x + 2 y = 4 to find y .
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The ordered pair is ( 5 4 , 1 2 ) .
Check the ordered pair in both equations.

4 x + 2 y = 4 4 ( 5 4 ) + 2 ( 1 2 ) = ? 4 5 1 = ? 4 4 = 4 6 x y = 8 6 ( 5 4 ) ( 1 2 ) = ? 8 15 4 ( 1 2 ) = ? 8 16 2 = ? 8 8 = 8
The solution is ( 5 4 , 1 2 ) .

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Solve the system by substitution. { x 4 y = −4 −3 x + 4 y = 0

( 2 , 3 2 )

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Solve the system by substitution. { 4 x y = 0 2 x 3 y = 5

( 1 2 , −2 )

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In [link] , it will take a little more work to solve one equation for x or y .

Solve the system by substitution. { 4 x 3 y = 6 15 y 20 x = −30

Solution

We need to solve one equation for one variable. We will solve the first equation for x .

.
Solve the first equation for x . .
Substitute 3 4 y + 3 2 for x in the second equation. .
Replace the x with 3 4 y + 3 2 . .
Solve for y . .
.
.

Since 0 = 0 is a true statement, the system is consistent. The equations are dependent. The graphs of these two equations would give the same line. The system has infinitely many solutions.

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Questions & Answers

the proudact of 3x^3-5×^2+3 and 2x^2+5x-4 in z7[x]/ is
anas Reply
?
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?
Manhwa Reply
44.19%
Scott
40.22%
Terence
44.2%
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
John Reply
168.50=R
Heather
john is 5years older than wanjiru.the sum of their years is27years.what is the age of each
achol Reply
46
mustee
j 17 w 11
Joseph
john is 16. wanjiru is 11.
Felix
27-5=22 22÷2=11 11+5=16
Joyce
where's the answers?
Ed Reply
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?
Maddy Reply
3
Christopher
18t+14t=96 32t=96 32/96 3
Christopher
show that a^n-b^2n is divisible by a-b
Florence Reply
What does 3 times your weight right now
Cherokee Reply
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
What is the lcm of 340
Kendra Reply
Yes
Cherokee

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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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