Solving Linear Systems of Equations by Elimination

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Example

Use elimination to find the solution of this system.

x - 3y = -17 First equation

-x + 8y = 52 Second equation

Solution

Add the two equations.

x - x	- +	3y 8y	= =	-	17 52
0x	+	5y	=		35

Simplify. The x-terms have been eliminated.

To solve for y, divide both sides by 5.

To find the value of x, substitute 7 for y in either of the original equations. Then solve for x.

= 35

= 7

We will use the first equation.

Substitute 7 for y.

Multiply.

Add 21 to both sides.

The solution of the system is (4, 7).

To check the solution, substitute 4 for x and 7 for y into each original equation. Then simplify.

In each case, the result will be a true statement.

The details of the check are left to you.

x - 3y

x - 3(7)

x - 21

= -17

= 4

In the two original equations in the previous example, the coefficients of x were opposites. Thus, when the equations were added, the x-terms were eliminated.

1x - 1x	- +	3y 8y	= =	-	17 52
		5y	=		35

When the coefficients of neither variable are opposites, we choose a variable. Then we multiply both sides of one (or both) equations by an appropriate number (or numbers) to make the coefficients of that variable opposites.

Note:

The Multiplication Principle of Equality enables us to multiply both sides of an equation by the same nonzero number without changing the solutions of the equation.