LEMMA

Lines & systems

Systems by elimination

ALEKS placement

Scale the equations until one variable has opposite coefficients, then add them.

What this covers

  • Line up like terms in both equations: x under x, y under y, plain numbers under plain numbers.
  • Pick the letter to cancel: the one with the smaller numbers, or one whose coefficients already match or are opposites.
  • Multiply one or both equations so that letter's coefficients are opposites. Multiply every term, the right side too.
  • Add the equations straight down. Each equation balances, so adding left to left and right to right keeps the balance.
  • Solve for the surviving letter, then substitute back for the other.
  • Check the pair in the equation you did not use for the back-substitution.

Reach for this when both equations are already in ax + by = c form and no variable is alone — especially when the coefficients are awkward numbers, since scaling avoids the fractions substitution would create. If one equation already reads y = something, substitution is faster.

Worked example

Worked example

Solve 2x + 3y = 12 and 4x - y = 10.

  1. 12x−3y=30Multiply the second equation by 3 so the y terms will cancel.
  2. 14x=42Add it to 2x + 3y = 12.
  3. x=3,y=2Divide, then substitute back.
Another worked example

Solve 3x + 2y = 8 and 2x − 3y = 1.

  1. 9x+6y=24Multiply every term of the first equation by 3, including the right side.
  2. 4x−6y=2Multiply every term of the second equation by 2. Now the y coefficients, 6 and −6, are opposites.
  3. 13x=26Add the two new equations. The y terms cancel.
  4. (x,y)=(2,1)Divide by 13 to get x = 2. Then 3(2) + 2y = 8 gives y = 1. Check: 2(2) − 3(1) = 1.

A common mistake

A mistake Lemma catches

From 4x−y=10

12x−3y=1012x−3y=30

“Multiplying an equation by 3 multiplies every term, the constant too: 3 · 4x = 12x, 3 · (−y) = −3y and 3 · 10 = 30. Leaving 10 as it was makes a different equation.”

Try one

Sample problem

Solve the system. Enter the ordered pair (x, y).

{x+y=−8x−y=−4​

Practice elimination free

No account, no email. Every line you type is checked by the same computer algebra system as the worked example above.

Where this fits

Before this