Solving Systems by Elimination

A system of equations is two equations that share the same variables, like x + y = 9 and x − y = 3. The solution of a system is the ordered pair (x, y) that makes both equations true. Elimination finds it without graphing: add or subtract the two equations so that one variable cancels. That leaves one equation with one unknown, which you already know how to solve. Then you go back for the other variable and check your pair in both equations. When the coefficients do not line up, you multiply an equation first to make them line up.

The big idea: make one variable disappear

Take x + y = 9 and x − y = 3. Stack them so like terms sit in columns, then add the left sides together and the right sides together:

(x + y) + (x − y) = 9 + 3

The y and the −y add to 0, so what is left is 2x = 12, and x = 6. Put x = 6 back into x + y = 9 to get 6 + y = 9, so y = 3. The solution is (6, 3).

Check it in both equations: 6 + 3 = 9 and 6 − 3 = 3.

Why is adding two equations allowed?

Adding two equations is the balance rule in disguise. Look at x + y = 9 and x − y = 3, and suppose (x, y) is a pair that makes both of them true.

• The balance rule says you may add the same amount to both sides of the first equation.
• The second equation says x − y and 3 are the same amount for that pair.
• So add x − y to the left side and 3 to the right side. These are equal amounts, so the first equation stays true: (x + y) + (x − y) = 9 + 3.

In short: if A = B and C = D, then A + C = B + D. Equal things added to equal things stay equal.

So the sum is true for every pair that solves both equations, and no solution is lost. Nothing false sneaks in either, because you keep one original equation next to the sum. The new pair of equations, x + y = 9 and 2x = 12, says exactly what the original pair said: subtracting the first equation from the sum rebuilds the second equation, so both systems have the same solutions.

Multiplying works the same way. Multiplying both sides of an equation by the same nonzero number gives an equation with the same solutions. (Multiplying by 0 is not allowed: it turns every equation into 0 = 0 and throws the information away.) Subtracting an equation is the same as multiplying it by −1 and then adding.

When is elimination convenient?

Elimination works on any system of two linear equations in two variables. It is quickest when:

• the equations are in standard form, Ax + By = C, so the x-terms line up, the y-terms line up, and the constants line up;
• one variable already has opposite coefficients (like 2y and −2y) or equal coefficients (like 3x and 3x).

A coefficient is the number multiplying a variable. An unwritten coefficient is 1, so x means 1x and −y means −1y.

If a variable is already alone, as in y = 3x − 1, substitution is often quicker. Elimination still works once you rewrite that equation in standard form. Graphing shows you the picture. All the methods give the same solution.

Adding when the coefficients are opposites

If a variable has opposite coefficients (they add to 0), add the equations: add the left sides, add the right sides, and combine like terms. That variable cancels, and you solve the equation that is left. Example 1 below does this with 2y and −2y.

Adding only helps when the coefficients are opposites. If they are equal, like 3y and 3y, adding gives 6y, and the variable is still there.

Subtracting when the coefficients are equal (watch the signs)

If a variable has the same coefficient in both equations, subtract one equation from the other, and that variable cancels.

The trap is the signs. You are subtracting the whole second equation, so the minus sign reaches every term on both sides, just like a minus sign in front of parentheses. For (3x + 5y) − (3x − 2y), the −2y becomes +2y, and a negative constant would become positive too:

(3x + 5y) − (3x − 2y) = 3x + 5y − 3x + 2y = 7y

If subtracting makes you nervous, multiply the second equation by −1 (every term) and then add. It gives the same result.

Making the coefficients match: multiply first

Often no variable has equal or opposite coefficients. Then multiply one or both equations by nonzero numbers so that one variable's coefficients become equal or opposite.

• Pick the variable that is easier to match. To match 2x and 5x, aim for 10, a common multiple of 2 and 5: multiply the first equation by 5 and the second by 2.
• If one coefficient is already a multiple of the other, multiply only one equation.
• Multiply every term, including the constant. Multiplying x − 3y = 6 by 4 gives 4x − 12y = 24, not 4x − 12y = 6.
• A negative multiplier is fine. Use it to make opposites, then add.

Then add or subtract as before. The other equation is untouched unless you chose to multiply it as well.

The full plan

1. Line up both equations in standard form, like terms in columns.
2. Adjust by multiplying one or both equations (every term, by a nonzero number) until one variable has opposite or equal coefficients. Skip this step if they already do.
3. Merge: add the equations if the coefficients are opposites, subtract if they are equal. Then solve the one-variable equation.
4. Back-substitute: put that value into one of the original equations and solve for the other variable.
5. Show it works: check the pair in both original equations, and write the answer as an ordered pair.

Back-substitute, then check in both equations

After you find one variable, put its value into either original equation to find the other. Choose the equation with the smaller numbers. Then write the answer as an ordered pair (x, y). Finding only x is half an answer.

Check the pair in both original equations. The equation you used to find the second value is satisfied by the way you found it (unless you slipped on the arithmetic there), so the other equation is what tests your elimination step. Check the originals, not the rewritten or multiplied versions, in case the slip happened while rewriting.

When both variables vanish: no solution or infinitely many

Sometimes both variables cancel at once. That happens when the left sides of the two equations are multiples of each other. What is left is a plain statement about numbers, so read it, as you did in the lesson on variables on both sides.

• False, like 0 = 17: no pair (x, y) can make both equations true, so the system has no solution. The lines are parallel (for lines that are not vertical: same slope, different y-intercepts). A simple case you can see by inspection: x + y = 5 and x + y = 8 cannot both be true, because x + y cannot equal 5 and 8 at the same time.
• True, like 0 = 0: the second equation says the same thing as the first, so both describe the same line. Every point on it solves both equations: infinitely many solutions.
• If a variable is left, you get exactly one solution.

Careful: 0 = 0 does not mean x = 0 and y = 0. “Infinitely many” is not a pair to plug in. It means every point on the line works.