1. Elimination Using Multiplication
Lesson 7-4
Elimination Using Multiplication

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4. Objectives
Solve systems of equations by using elimination with multiplication
Determine best method for solving systems of equations

5. Vocabulary
none new

6. Solve Systems of Equations: Elimination
Sometimes we can multiply two sets of equations by a constant and add them together to eliminate a variable
Example: Solve 2x + 4y = 20 and -3x + 8y = 26
2x + 4y = (equation one; 4y  2 = 8y)
 x + 8y = (equation one 2)
x + 8y = (equation two) (-#) is a positive
7x = Eliminate y by subtracting
x = Divide both sides by 7
2(2) + 4y = Sub x= into equation one
4y = Simplifying
y = Divide both sides by 4

7. Example 1 Use elimination to solve the system of equations.
Multiply the first equation by –2 so the coefficients of the y terms are additive inverses. Then add the equations.
Multiply by –2.
Add the equations.
Divide each side by –1.
Simplify.

8. Example 1 cont
Now substitute 9 for x in either equation to find the value of y.
First equation
Simplify.
Subtract 18 from each side.
Simplify.
Answer: The solution is (9, 5).

9. Example 2 Use elimination to solve the system of equations.
Method 1 Eliminate x.
Multiply by 3.
Multiply by –4.
Add the equations.
Divide each side by 29.
Simplify.

10. Example 2 cont Now substitute 4 for y in either equation to find x.
First equation
Simplify.
Subtract 12 from each side.
Simplify.
Divide each side by 4.
Simplify.
Answer: The solution is (–1, 4).

11. Example 2 – Another Way Method 2 Eliminate y. Multiply by 5.
Add the equations.
Divide each side by 29.
Simplify.

12. Example 2 – Another Way cont
Now substitute –1 for x in either equation.
First equation
Simplify.
Add 4 to each side.
Simplify.
Divide each side by 3.
Simplify.
Answer: The solution is (–1, 4), which matches the result obtained with Method 1.

13. Example 3
Determine the best method to solve the system of equations. Then solve the system.
For an exact solution, an algebraic method is best.
Since neither the coefficients for x nor the coefficients for y are the same or additive inverses, you cannot use elimination using addition or subtraction.
Since the coefficient of the x term in the first equation is 1, you can use the substitution method. You could also use the elimination method using multiplication.

14. Example 3 cont The following solution uses substitution.
First equation
Subtract 5y from each side.
Simplify.
Second equation
Distributive Property
Combine like terms.
Subtract 12 from each side.
Simplify.

15. Example 3 cont Simplify. Divide each side by –22. Simplify.
First equation
Simplify.
Subtract 5 from each side.
Simplify.
Answer: The solution is (–1, 1).

16. Example 4
Transportation A fishing boat travels 10 miles downstream in 30 minutes. The return trip takes the boat 40 minutes. Find the rate of the boat in still water.
Let b = the rate of the boat in still water. Let c = the rate of the current. Use the formula rate  time = distance, or rt = d. Since the rate is miles per hour, write 30 minutes as ½ hour and 40 minutes as ⅔ hour. 10
Upstream
Downstream d t r
This system cannot easily be solved using substitution. It cannot be solved by just adding or subtracting the equations.

17. Example 4 cont
The best way to solve this system is to use elimination using multiplication. Since the problem asks for b, eliminate c.
Multiply by .
Multiply by .
Add the equations.
Multiply each
side by
Simplify.
Answer: The rate of the boat is 17.5 mph.

18. Solving Systems of Equations
Three methods for solving systems of equations:
Graphing (from 7.1)
Substitution (from 7.2)
Elimination (from 7.3 and 7.4)
using addition,
subtraction or
multiplication

19. Summary & Homework Summary: Homework:
Multiplying one equation by a number or multiplying a different number is a strategy that can be used to solve systems of equations by eliminations
Three methods for solving systems of equations:
Graphing
Substitution
Elimination (using addition, subtraction or multiplication)
Homework:
Pg even
There is a graph