Solve systems of linear equations by elimination

Warm-up

Three quick questions from earlier lessons.

Discover

$x + y = 10$ and $x - y = 4$ . Add the two equations. What happens?

$y$ disappears: $2x = 14$
$x$ disappears

Key idea

Elimination: add or subtract whole equations.

Adding or subtracting equations keeps the solution of the system. Elimination: make one variable's coefficients equal (multiply an equation if needed), then subtract (or add if they are opposite) so that variable disappears.

Worked example

$2x + 5y = 19$ and $2x + y = 11$ .

4y = 8

Subtract: 2x cancels.

y = 2

Divide.

x = 4.5

2x + 2 = 11.

Your turn, guided

Your turn: $x + y = 12$ and $x - y = 2$ . Find $x$ .

x = 7

Add: 2x = 14.

Practice ★

Solve the system $x + y = 15$ and $x - y = 3$ . $x = \;?$
Solve the system $x + y = 3$ and $x - y = -1$ . $x = \;?$

Solve the system $3x + 4y = 30$ and $3x + 2y = 18$ . $y = \;?$

Watch out!

Practice ★★

To solve $x + 7y = 20$ and $9x + y = 20$ by elimination, you multiply the first equation by a number and subtract the second, so that $x$ disappears. Which number?
To solve $x + 5y = 20$ and $9x + y = 27$ by elimination, you multiply the first equation by a number and subtract the second, so that $x$ disappears. Which number?

Boss problem. A school play sold 33 tickets: adult tickets at 9 dollars and child tickets at 7 dollars, for 287 dollars in all. Let $x$ be the number of adult tickets and $y$ the number of child tickets.

Exit ticket

Three questions to check you have got it.

Solve the system $x + y = 15$ and $x - y = 3$ . $x = \;?$
Solve the system $x + y = 3$ and $x - y = -1$ . $x = \;?$
Solve the system $x + y = 8$ and $x - y = -6$ . $x = \;?$

Summary card

Elimination

Equal coefficients: subtract. Opposite coefficients: add.

Watch out: multiply first when no coefficients match.

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