If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

# Systems of equations with elimination (and manipulation)

In some cases, we need to slightly manipulate a system of equations before we can solve it using the elimination method. See how it's done in this video. Created by Sal Khan.

## Want to join the conversation?

• Is elimination the only way to solve linear equations
• One may find it easier to use matrices when he is faced with crazy equations including five or so variables and five or so complicated equations. Otherwise, substitution and elimination are your best options. Graphing, unless done extremely precisely, may lead to error.
• how do you eliminate negative numbers?
• You just add the number you do the opposite of the sign
Rule: + x - = -
- x + = -
• I don't understand why if you subtract negative 15 from 5 you don't get 20....?
• He is adding, not subtracting. That is why he had to make the numbers negative in order to cancel them out. 5x+(-15x)=-10x. Adding a -15 is like subtracting a +15.
• You know the second equation couldn't he just multiply that by 5x? Did it have to be negative 5?
• Mye,
He used a negative 5 so he could just add the two equations and the 10y and -10y become 0y and eliminate the y.
And you are correct. He could have just used a 5 instead of a -5, but then he would have had to subtract the equations instead of adding them. That would work the same way and you get the same answer.

I hope that helps.
• When you say ' 5 is the same as 20/4' dont understand how ??
• 5 is the same as 20/4 because 20/4 is 20 divided by 4 which equals 5
• At where did the -5 come from?
• Sal chose to multiply both sides of the bottom equation by -5. I know, I know, you want to know why he decided to do that. Well he wanted at least one term with a variable in each equation to be the same size but opposite in sign. Since the top equation was
5x-10y =15 and the bottom equation was 3x - 2y = 3, he recognized that by multiplying both sides of the bottom equation by -5 he could get the "y" terms in each equation to be the same size (10) but opposite in sign ... that way if he added the two equations together, he would "ELIMINATE" the "y" term and then he would just have to solve for x.
• Do the answers multiply back to the original if factored?
• how would you figure out what x and y are if the equation cancels both out
ex: 3x + 2y = 18
6x + 4y = 8
• With this problem, there is no solution. If you multiply 3x + 2y = 18 by -2 (I chose -2 so when you add the equations together, variables cancel out), you get -6x - 4y = -36. When you add -6x - 4y = -36 and 6x + 4y = 8, you get 0 on the left side of the equation and -28 on the right side. Since 0 = -28 is untrue, the answer to this system of equations is "no solution."