Solving a System by Elimination

If one equation has +y and the other −y, adding them removes y entirely.

When the signs match

Two equations both with +y are subtracted instead. Either way the y goes.

When nothing matches

Multiply one equation through so a pair of coefficients agrees. 2x and 3x become 6x and 6x.

Same ending as substitution

Solve the one remaining letter, then put it back to find the other.

Add or subtract to remove a variable

If one equation has +3y and the other −3y, adding them removes y entirely and leaves one equation in x. Elimination works by arranging for one variable's coefficients to cancel.

Scale first when they do not match

For 2x + 3y = 12 and 3x + 2y = 13, multiply the first by 3 and the second by 2 to make the x terms match, then subtract. Multiplying a whole equation by a number keeps it true — every term, both sides.

Add when signs differ, subtract when they match

Coefficients of +3y and −3y cancel on addition; two +3y terms cancel on subtraction. Subtracting a whole equation means subtracting every term, and a dropped sign there is the standard elimination error.

Choosing between the two methods

Substitution suits systems where a variable is already isolated. Elimination suits systems in standard form with matching or easily matched coefficients. Both always work, so pick whichever the numbers make easier.