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Lesson 7-4
Elimination Using Multiplication
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Objectives
• Solve systems of equations by using
elimination with multiplication
• Determine best method for solving systems of
equations
Vocabulary
• none new
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 = 20
2
4x + 8y = 40
-3x + 8y = 26
7x
= 14
x=2
2(2) + 4y = 22
4y = 18
y=4
(equation one; 4y  2 = 8y)
(equation one 2)
(equation two)
-(-#) is a positive
Eliminate y by subtracting
Divide both sides by 7
Sub x= into equation one
Simplifying
Divide both sides by 4
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.
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).
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.
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).
Example 2 – Another Way
Method 2 Eliminate y.
Multiply by 5.
Multiply by 3.
Add the equations.
Divide each side
by 29.
Simplify.
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.
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.
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.
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).
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.
r
t
d
Downstream
10
Upstream
10
This system cannot easily be solved using substitution. It
cannot be solved by just adding or subtracting the equations.
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.
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
Summary & Homework
• Summary:
– 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 391 14-38 even