How can I solve a two-equation system by elimination?

First write each equation with the variable terms together and the constant on the other side, in the same order. This is often called standard form; for example, write both with the x-term first and the y-term second. Then compare the coefficients of x with each other and the coefficients of y with each other. You want one variable’s coefficients to be opposites, such as +3 and −3, so that adding the equations removes that variable. If they already are opposites, you can add right away. If they are the same instead, subtracting one equation from the other can cancel that variable. Elimination relies on doing equivalent operations to equations, so when you multiply an equation, multiply every term on both sides by the same number. [S1]

Try this example:

2x + 3y = 13 4x − y = 5

The y-coefficients, 3 and −1, are easy to turn into opposites. Multiply the entire second equation by 3:

12x − 3y = 15

Keep the first equation unchanged, then add:

2x + 3y = 13 12x − 3y = 15 14x = 28

The +3y and −3y cancel, leaving x = 2. The useful choice was to make the y-coefficients opposites; it was not necessary to change both equations. Multiplication changes the equation’s appearance, not the set of values that satisfies it, provided every term is multiplied consistently. Once one variable is eliminated, solve the one-variable equation before trying to find the other value. This is the core elimination sequence: arrange, match coefficients, add or subtract, and solve. [S1]

Use the value you found in one of the original equations, rather than an altered equation if that feels clearer. In the example, substitute x = 2 into 2x + 3y = 13:

2(2) + 3y = 13 4 + 3y = 13 3y = 9, so y = 3.

The proposed solution is the ordered pair (2, 3). Check it in both original equations: 2(2) + 3(3) = 4 + 9 = 13, and 4(2) − 3 = 8 − 3 = 5. Both statements are true, so the pair works for this system. Checking both originals matters: a value that satisfies only one equation is not enough to establish a solution to the system. [S1]

If the result looks unexpected, check that you multiplied every term consistently and that the signs and addition are correct. For a typical elimination walkthrough where one variable remains, use this template:

1. Put both equations in the same standard form.
2. Choose x or y to eliminate; make its coefficients opposites by multiplying whole equations if needed.
3. Add the equations and solve the remaining one-variable equation.
4. Substitute that value into an original equation to find the other variable.
5. Substitute the ordered pair into both original equations and check each equality. [S1]

Keep the multiplied equation visible while adding, and write down the cancellation explicitly. Those small habits make sign or multiplication errors easier to spot.