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### Course: Integrated math 1 > Unit 6

Lesson 2: Solving systems of equations with substitution# Substitution method review (systems of equations)

The substitution method is a technique for solving a system of equations. This article reviews the technique with multiple examples and some practice problems for you to try on your own.

## What is the substitution method?

The substitution method is a technique for solving systems of linear equations. Let's walk through a couple of examples.

### Example 1

We're asked to solve this system of equations:

The second equation is solved for $x$ , so we can substitute the expression $-y+3$ in for $x$ in the first equation:

Plugging this value back into one of our original equations, say $x=-y+3$ , we solve for the other variable:

The solution to the system of equations is $x=-3$ , $y=6$ .

We can check our work by plugging these numbers back into the original equations. Let's try $3x+y=-3$ .

Yes, our solution checks out.

### Example 2

We're asked to solve this system of equations:

In order to use the substitution method, we'll need to solve for either $x$ or $y$ in one of the equations. Let's solve for $y$ in the second equation:

Now we can substitute the expression $2x+9$ in for $y$ in the first equation of our system:

Plugging this value back into one of our original equations, say $y=2x+9$ , we solve for the other variable:

The solution to the system of equations is $x=-2$ , $y=5$ .

*Want to learn more about the substitution method? Check out this video.*

## Want to join the conversation?

- I am curious if there are times when either the elimination method or the substitution method would be more appropriate, and or if there would be times when only one way or the other would work. Thank you for the advice in advance!(58 votes)
- Yes, both equations will always work, but yes, at times it is more logical to use one over the other. for instance:

x-4y=6

x+4y=12

we see here that elimination would best fit:

x+x+4y-4y=6+12

or:

x=2y-3

7x+3y-81=23

substitute (2y-3) for x.

Also, once you have a single placeholder, put it ito quadratic form (ax^2+bx+c=0)

and use the quadratic formula:

x=(-b±sqrt(b^2-4ac))/2a(41 votes)

- This is pretty challenging not gonna lie(58 votes)
- If you practice a lot and have a good teacher, it'll get less challenging.(19 votes)

- how do I solve y=2x-1 and y=3x+2 using the substitution method(8 votes)
- You would set them equal to each other because they both equal "y": 2x-1=3x+2(11 votes)

- are there any easy tips or tricks i can use to remember this?(10 votes)
- Lol this is actually kind of simple, though it takes a different mindset. I like to think of the first equation as the problem I'm trying to solve and the second as a hint. See the second one says what x equals even if it doesn't necessarily give you the answer. Because it doesn't give the answer for y, we have to replace the x with the second equation to be able to solve for it. After solving for y, you would plug y into one of the two original equations and solve for x the way you did for y. After you solve for x, you would have the two answers you needed, x and y :) (I hope this was helpful, sorry if it wasn't :[ ...)(10 votes)
- I know! Once I got how to solve the equations it was so fun just getting through it.(3 votes)

- Practice Solutions:

First system of equations:

1st equation: -5x + 4y = 3

2nd equation: x = 2y - 15

Substitute for x:

-5(2y - 15) + 4y = 3

Solve for y:

-10y + 75 + 4y = 3

-6y = -72

y = 12

Solve for x by substituting in one of the equations:

x = 2y - 15

x = 2(12) - 15

x = 9

Check your solution by substituting for x and y in the first equation:

-5x + 4y = 3

-5(9) + 4(12) = 3

-45 + 48 = 3

3 = 3 √

Second system of equations:

5x - 7y = 58

y = -x + 2

Subsitute for y:

5x - 7(-x + 2) = 58

Solve for x:

5x + 7x - 14 = 58

12x - 14 = 58

12x = 58 + 14

x = 72/12

x = 6

Solve for y by substituting in one of the equations:

y = -x + 2

y -(6) + 2

y = -4

Check your solution by substituting for x and y in the first equation:

5x - 7y = 58

5(6) - 7(-4) = 58

30 + 28 = 58

58 = 58 √

I hope this could help someone. It seems like a lot, but once you understand the steps and get the hang of it, it will become pretty quick!(10 votes) - why does math exist?(4 votes)
- Think about everything you do day-day that uses numbers. All those tasks need math. Business use math. Video games use math. Engineering uses math. Forecasting weather uses math. Architecture and construction use math. Sports and other games use math. Medicine uses math. There are many other things that use math.(9 votes)

- who is up for quiz one(9 votes)
- If i may ask,could anyone help me in this equation:

3y+2x=7

y-3x=6

Please answer the question in with explaination.thanks🙏🏽🥺😊(4 votes)- I'll get you started.

Take the 2nd equation and add 3x to both sides to isolate "y".

y = 3x+3

Now, substitute this value of "y" into the first equation.

3(3x+6) + 2x = 7

You can now solve for "x".

Once you have the value of "x", use it to calculate the value of "y".

Hope this helps. Comment back if you have questions.(8 votes)

- In example 2, when it says we have to solve for x or y, how do I get -2x+y=9 into slope intercept form? And also, how did we get rid of the negative once it was in slope intercept form?

Thanks(5 votes)- You cannot get rid of negatives if they are there, so let the signs work themselves out. To isolate the y, all you need to do is add 2x on both sides to get y=2x+9.(6 votes)