Elimination — and Why Adding a Multiple Is Legal

In this lesson you will: Add the two equations, or a multiple of one to the other, so that one variable cancels; Scale an equation by multiplying every term, both sides, by the same number; Justify the method: the new system has exactly the same solutions as the old one.

Stack them and add

Line the variables up in columns. If the coefficients of one variable are opposites, adding the two equations makes that variable disappear and leaves an equation you can solve on sight.

For x+y=10 and x-y=4, the y terms are +y and -y. Adding gives 2x=14, so x=7, and then 7+y=10 gives y=3.

When nothing cancels yet, scale first

Most systems do not arrive with opposite coefficients. Manufacture them. Multiply one equation — every term of it, both sides — by whatever number makes a pair of coefficients opposite.

In 3x+y=14 and 2x-3y=2, multiplying the first equation by 3 gives 9x+3y=42. Now the y terms are +3y and -3y. Adding gives 11x=44, so x=4 and y=2.

Why this is allowed — the A.REI.5 argument

Elimination replaces one of your equations with a brand-new one that was nowhere in the original problem. That should bother you slightly. Why is the answer to the new system the answer to the old one?

Suppose the pair (a,b) satisfies both E_1 and E_2. Then the left side of E_1 equals its right side, and k times the left side of E_2 equals k times its right side. Adding equals to equals keeps equality, so (a,b) satisfies E_1+kE_2 as well. Every old solution survives.

Now run it backwards. Suppose (a,b) satisfies E_2 and the new equation E_1+kE_2. Subtract k times E_2 from the new equation and you are back at E_1, so (a,b) satisfies E_1 too. No new solutions were invented.

Both directions together say the two systems have exactly the same solution set. The move is reversible, and that is the whole justification.

Common mistake — Multiplying the left side and forgetting the right

To eliminate y from 4x-y=9 and x+3y=7, a student multiplies the first equation by 3 and writes 12x-3y=9.

The eye is on the coefficients, because the coefficients are what have to match. The right side is just sitting there, already tidy, and it gets left alone.

But an equation is a statement that two quantities are equal. Tripling one of them and not the other breaks that statement. The correct line is 12x-3y=27, and everything after the missed 27 is wrong however careful the rest of the work is.

How to catch it: Say the multiplier out loud as you write each term, right side included: \"three times 4x, three times -y, three times 9.\" Or check the new equation with a point you know is on the old line — (3,3) satisfies 4x-y=9, and 12(3)-3(3)=27, not 9.