Solving Systems of Equations
Using Elimination
Solving Systems by Elimination
Elimination Using Addition
Example 1 Elimination Using Addition
Use elimination to solve each system of equations.
-4x + 3y = 17
4x + 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.
-4x + 3y = 17 Write the equations in column form and add.
(+)4x + y = 3
4y = 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.
4x + y = 3 Second equation
4x + 5 = 3 Replace y with 5.
4x + 5 – 5 = 3 – 5 Subtract 5 from each side.
4x = -2 Simplify.
4x= -2 Divide each side by 4.
x = - (1/2) Simplify.
The solution is (-1/2, 5)
Review
Use When....?
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.
7a + 3b = 3
2a + 3b = 18
Since the coefficients of the b-terms, 3 and 3, are the same, you can eliminate the b-terms by subtracting the equations.
7a + 3b = 3 Write the equations in column form and subtract.
(-) 2a + 3b = 18
5a = -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.
2a + 3b = 18 Second equation
2(-3) + 3b = 18 a = -3
-6 + 3b = 18 Simplify.
3b = 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 + 3y = -4 Multiply by –3. -x – 9y = 12
x + 2y = 9 (+) x + 2y = 9
-7y = 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 + 2y = 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.
5x – 7y = -2
-4x + 6y = 4
Method 1 Eliminate x.
5x – 7y = -2 Multiply by 4. 20x – 28y = -8
-4x + 6y = 4 Multiply by 5. (+) -20x + 30y = 20
2y = 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.
5x – 7y = -2 First equation
5x – 7(6) = -2 y = 6
5x – 42 = -2 Simplify.
5x – 42 + 42 = -2 + 42 Add 42 to each side.
5x = 40 Simplify.
x = 8 Divide each side by 5 and simplify.
The solution is (8, 6).