Algebra · Simultaneous equations
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Simultaneous equations by elimination
Why two unknowns need two equations, why adding or subtracting whole equations is legal, and the sign rule that decides which of the two kills a variable.
Algebra · Simultaneous equations
Simultaneous equations by elimination
Why two unknowns need two equations, why adding or subtracting whole equations is legal, and the sign rule that decides which of the two kills a variable.
Why it works
One equation with two unknowns has endless solutions — is satisfied by whole families of pairs. A SECOND equation cuts the family down to a single pair: the values that satisfy both at once. That's what "simultaneous" means.Adding equations is legal because both sides balance. If and , then adding left sides and right sides keeps truth: . And look what happened — and cancel:
then substitute back into either equation: , so . Elimination is engineered cancellation.
The sign rule: matching coefficients decide add or subtract.
- Coefficients equal, opposite signs ( and ) → ADD.
- Coefficients equal, same sign ( and ) → SUBTRACT.
Scale first if nothing matches. In and , no coefficients match — so multiply the second whole equation (right side included!) by 3: . Now subtract: , , . Sometimes BOTH equations need scaling: for and , take and to match against , subtract, and .
Finish properly. Finding is half the job: substitute back for the second unknown, then check the pair in the equation you didn't just use — the only check that can actually catch an error.
Subtracting is where the marks die. style work breeds sign slips; write the subtraction out term by term, or choose the add-route by scaling with opposite signs when you can.