XMocks · SAT Math · Systems

Use symmetry to solve a system by sum and difference

Some systems have the same two coefficients in reversed order. Instead of eliminating a variable with a long multiplication step, add the equations to find the sum of the variables and subtract them to find their difference.

Recognize the pattern

Consider ax + by = m and bx + ay = n. Adding the equations gives (a + b)(x + y) = m + n. Subtracting the first equation from the second gives (b − a)(x − y) = n − m. When these coefficients are nonzero, the equations provide the sum and difference directly.

This method is ordinary elimination viewed through the coefficient pattern. It does not change the solution; it helps you choose operations that reveal useful information quickly.

Worked example: solve and verify

Solve the system 3x + 5y = 28 and 5x + 3y = 36.

  1. Add the equations: 8x + 8y = 64, so x + y = 8.
  2. Subtract the first equation from the second: 2x − 2y = 8, so x − y = 4.
  3. Add the simpler equations: 2x = 12, giving x = 6. Then 6 + y = 8, so y = 2.

Check the original system

3(6) + 5(2) = 18 + 10 = 28 and 5(6) + 3(2) = 30 + 6 = 36. Both equations are true, so the solution is (6, 2).

Why addition and subtraction help

Adding removes the distinction between x and y in the left side because both receive the combined coefficient a + b. Subtracting keeps the difference because the x-terms and y-terms move in opposite directions. After that, the original two-variable problem becomes a smaller system involving x + y and x − y.

Watch the order of subtraction. Reversing the equations changes the signs of both differences, but it does not change the final solution if the arithmetic is consistent. A quick substitution into both original equations is a reliable sign check.

Practice: use the two relationships

Solve 4x + 7y = 41 and 7x + 4y = 47. Find x + y and x − y before solving for each variable.

Show the worked answer

Adding gives 11x + 11y = 88, so x + y = 8. Subtracting the first equation from the second gives 3x − 3y = 6, so x − y = 2. Add these relationships to get 2x = 10, hence x = 5 and y = 3. Check: 4(5) + 7(3) = 41 and 7(5) + 4(3) = 47.

Practice and next steps

Try a new pair of equations with reversed coefficients. First predict whether the sum or difference will be simpler, then verify your answer in both equations.

This independently prepared XMocks lesson is not affiliated with or endorsed by the College Board.