Solving Systems by Elimination: notes and common mistakes

Key ideas

• Elimination adds or subtracts the two equations so that one variable cancels, leaving one equation in one variable.
• Adding equations is allowed because equal amounts added to both sides keep an equation true: if A = B and C = D, then A + C = B + D. Keep one original equation next to the new one and the system has the same solutions.
• Add when a variable's coefficients are opposites. Subtract when they are equal. Otherwise multiply one or both equations by nonzero numbers first (every term, including the constant).
• Subtracting an equation means subtracting every term on both sides, so every sign in that equation flips.
• Back-substitute to find the other variable, then check the ordered pair in both original equations.
• If both variables cancel, a false statement means no solution and a true statement means infinitely many solutions.

Steps

1. Write both equations in standard form Ax + By = C, with like terms in columns.
2. Choose a variable to eliminate. Multiply one or both equations (every term, never by 0) until its coefficients are opposites or equal.
3. Add the equations if the coefficients are opposites. Subtract if they are equal. Solve the equation that is left.
4. Put that value into an original equation and solve for the other variable.
5. Write the solution as an ordered pair (x, y).
6. Check the pair in both original equations. If both variables vanished, read the leftover statement instead.

Formulas

• Standard form: Ax + By = C
• Adding equations: If A = B and C = D, then A + C = B + D
• Subtracting equations: If A = B and C = D, then A − C = B − D
• Multiplying an equation: A = B and kA = kB have the same solutions when k ≠ 0

Watch out

• Add only when the coefficients are opposites, and subtract only when they are equal. Adding 3y and 3y gives 6y, and the variable stays.
• When you subtract an equation, subtract every term, including the constant: in (3x + 5y) − (3x − 2y), the y-terms give 5y − (−2y) = 7y.
• When you multiply an equation, multiply every term, including the constant on the right. Never multiply by 0.
• Do not stop at x. The solution is an ordered pair, so find both numbers.
• Check in both original equations, not in the rewritten ones.
• A leftover 0 = 0 does not mean x = 0 and y = 0. It means infinitely many solutions.

Why it works

• Why can you add equations? If (x, y) makes both equations true, the second equation says two expressions are equal, so adding them to the two sides of the first equation keeps it true. Equal amounts added to both sides of an equation keep it balanced.
• Why does a false leftover mean no solution? If some pair solved both equations, it would also solve the sum, but the leftover says something like 0 = 17, which is never true. So no pair can solve both.
• Why does a true leftover mean infinitely many solutions? The equations say the same thing, so every pair that solves one also solves the other. A line has infinitely many points.
• Why must you multiply every term? Multiplying both sides by the same nonzero number keeps the equation balanced. Skipping the constant would change only one side.