# Solving Systems of Equations

### Using Elimination

## Elimination Using Addition

** **

__Example 1__** Elimination Using Addition**

** **

**Use elimination to solve each system of equations.**

** -4 x + 3y = 17**

** 4 x + y = 3**

Since the coefficients of the *x*-terms, -4 and 4, are additive inverses, you can eliminate the *x*-terms by adding the equations.

-4*x* + 3*y* = 17 Write the equations in column form and add.

__(+)4 x + y = 3__

4*y* = 20 Notice the *x* variable is eliminated.

4y= 20 Divide each side by 4.

*y* = 5 Simplify.

Now substitute 5 for *y* in either equation to find the value of *x*.

4*x* + *y* = 3 Second equation

4*x* + 5 = 3 Replace *y* with 5.

4*x* + 5 – 5 = 3 – 5 Subtract 5 from each side.

4*x* = -2 Simplify.

4x= -2 Divide each side by 4.

*x* = - (1/2) Simplify.

The solution is (-1/2, 5)

## Review

## Use When....? All systems can be solved in more than one way. For some systems, some methods may be better than others. | ## Another way to look at things. | ## Don't Forget! |

## Elimination Using Subtraction

## Hint: same term, you subtract

__Example 2__ Elimination Using Subtraction

** **

**Use elimination to solve the system of equations.**

** **7*a* + **3 b** = 3

2*a* +** 3 b** = 18

Since the coefficients of the *b*-terms, 3 and 3, are the same, you can eliminate the *b*-terms by subtracting the equations.

7*a* + 3*b* = 3 Write the equations in column form and subtract.

**(-)** 2*a* + 3*b* = 18

5*a* = -15 The variable *b* is eliminated.

5a= -15 Divide each side by 5.

*a* = -3 Simplify.

Now substitute –3 for *a* in either equation to find the value of *b*.

2*a* + 3*b* = 18 Second equation

2(-3) + 3*b* = 18 *a* = -3

-6 + 3*b* = 18 Simplify.

3*b* = 24 Add 6 to each side and simplify.

3b= 24 Divide each side by 3.

*b* = 8

The solution is (-3, 8).

## Using Multiplication

__Example 1__**Multiply One Equation to Eliminate**

** Use elimination to solve the system of equations.**

** x + 3y = -4**

** x + 2y = 9**

Multiply the first equation by –3 so the coefficients of the *x*-terms are additive inverses. Then add the equations.

*x* + 3*y* = -4 Multiply by –3. -*x* – 9*y* = 12

*x* + 2*y* = 9 __(+) x + 2y = 9__

-7*y* = 21 Add the equations.

= Divide each side by –7.

*y* = -3

Now substitute –3 for *y* in either equation to find the value of *x*.

*x* + 2*y* = 9 Second equation

* x* + 2(-3) = 9 *y* = -3

* x *– 6 = 9 Simplify.

* x* = 15 Add 6 to each side and simplify.

The solution is (15, -3).

## Using Multiplication

__Example 2__ Multiply Both Equations to Eliminate

** **

**Use elimination to solve the system of equations.**

** 5 x – 7y = -2**

** -4 x + 6y = 4**

** Method 1 **Eliminate *x*.

** **5*x* – 7*y* = -2 Multiply by 4. 20*x* – 28*y* = -8

-4*x* + 6*y* = 4 Multiply by 5. __(+) -20 x + 30y = 20__

2*y* = 12 Add the equations.

2y= 12 Divide each side by 2.

*y* = 6 Simplify.

Now substitute 6 for *y* in either equation to find the value of *x*.

5*x* – 7*y* = -2 First equation

5*x* – 7(6) = -2 *y* = 6

5*x* – 42 = -2 Simplify.

5*x* – 42 + 42 = -2 + 42 Add 42 to each side.

5*x* = 40 Simplify.

*x* = 8 Divide each side by 5 and simplify.

The solution is (8, 6).