Two friends stop at a café. Maya buys 3 hot chocolates and 2 bagels for $13. Jonah buys 5 hot chocolates and 2 bagels for $19. Neither receipt shows the price of a single item, but you can still work it out. Both orders include the same 2 bagels, so the only difference between them is that Jonah bought 2 more hot chocolates, and he paid $6 more. So 2 hot chocolates cost $6, and one costs $3. That idea, comparing two equations so that one variable cancels out, is the elimination method. In algebra you write the comparison as subtracting one equation from the other.
Why adding or subtracting equations works
An equation says that its two sides are equal amounts. If and , then and : adding equal amounts to equal amounts keeps things equal. So you may add (or subtract) the left sides of two equations and the right sides, and the result is still a true equation at the solution. The goal is to choose the operation that makes one variable disappear. If a variable has opposite coefficients, like and , add the equations. If it has the same coefficient, like and , subtract them. Here is a system with opposite -coefficients:
When the coefficients do not match: multiply first
Most systems do not arrive with matching coefficients, so you create them. Multiplying every term of an equation by the same non-zero number gives an equivalent equation: it describes the same line. In and , multiply the first equation by to get . Now the -coefficients are and , so adding gives and ; then gives . Sometimes you must multiply both equations. In and , no coefficient is a multiple of its partner. To eliminate , aim for the lowest common multiple of and , which is : multiply the first equation by and the second by , as the figure shows.
- Write both equations in the same form, , with like terms lined up in columns.
- Choose a variable to eliminate. Multiply one or both equations so that its coefficients are equal or opposite.
- Add the equations if the coefficients are opposite; subtract if they are equal. Combine every column, including the constants.
- Solve the resulting one-variable equation.
- Back-substitute into either original equation to find the other variable.
- Check the ordered pair in both original equations.
Choosing a method
| Method | Works best when | Watch out for |
|---|---|---|
| Graphing | Both lines are easy to draw ( or whole-number intercepts) and you want a picture or an estimate | Answers that are not whole numbers can only be estimated |
| Substitution | One equation is already solved for a variable, or a variable has coefficient or | Fractions if no coefficient is 1 or −1; forgotten brackets |
| Elimination | Both equations are in standard form , especially when no coefficient is or | Sign errors when subtracting; multiplying only part of an equation |
Elimination reveals the special cases the same way substitution does. Solve and : multiply the first equation by to get , then subtract the second. Every term cancels and you get , a true statement, so the equations describe the same line and there are infinitely many solutions. Change the second equation to and the same steps give , a false statement, so the lines are parallel and there is no solution.