Solving Simultaneous Equations by Elimination

Eliminate one variable by adding or subtracting the equations, solve for the other, then check the pair in both originals.

Take the pair x + y = 7 and x - y = 1. The y coefficients are +1 and -1, so adding the two equations eliminates y: 2x = 8, giving x = 4.

Back-substitute x = 4 into the first equation: 4 + y = 7, so y = 3. The pair is (4, 3).

Check the pair in both originals: 4 + 3 = 7 and 4 - 3 = 1. Both hold, so the solution is confirmed; checking in only one equation is the classic way to lose the final mark.

If neither variable has matching coefficients, multiply one equation so that they match, then add or subtract exactly as before. The routine is identical once the coefficients line up.

Substitution does the same job: rearrange the first equation to y = 7 - x and replace y in the second, giving x - (7 - x) = 1, so 2x = 8 again. Use whichever route needs fewer lines; the quadratic page uses the same one-operation-per-line layout, and the discriminant page shows the same habit of deciding before solving.

• add when the matching coefficients have opposite signs
• subtract when the matching coefficients have the same sign
• multiply one equation first if no coefficients match
• always check the pair in both original equations

Further reading

• How to Solve a Quadratic Equation
• What the Discriminant Tells You