Simultaneous Equations by Elimination

Updated September 26, 2026•Algebra & Equations

Solve two linear equations by scaling and elimination, with sign control, substitution checks and efficient setup.

What elimination does

Two equations can describe the same pair of unknowns. Elimination creates a new equation containing only one unknown by making one set of coefficients equal and opposite.

When coefficients already match

Example:

2x + y = 11 3x − y = 9

Add the equations. The +y and −y cancel.

5x = 20 x = 4

Substitute into 2x + y = 11:

8 + y = 11 y = 3

Solution: x = 4, y = 3.

When you need to scale an equation

If the coefficients do not match, multiply one or both entire equations. For 2x+3y=13 and x+2y=8, doubling the second equation makes the x-coefficients both 2.

2x + 3y = 13 2x + 4y = 16 subtract first from second → y = 3 x + 2(3) = 8 → x = 2

Decide the sign before you combine the equations

If the target coefficients are opposite signs, add. If they are the same sign, subtract. Writing the equations vertically reduces sign errors.

One solution has to satisfy both lines

A pair is only correct if it satisfies both equations. Substitution into both originals catches many arithmetic mistakes.

Choose the cancellation that keeps the numbers small

Choose the variable that needs the smallest multipliers to eliminate. The method is the same, but cleaner numbers reduce errors.

Try one yourself

For x + y = 10 and x − y = 2, what is x?

Keep going

For another problem using related ideas, return to the Algebra & Equations page.