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 (equation one; 4y 2 = 8y)

2 4x + 8y = 40 (equation one 2)

- -3x + 8y = 26 (equation two) -(-#) is a positive

7x = 14 Eliminate y by subtracting

x = 2 Divide both sides by 7

2(2) + 4y = 22 Sub x= into equation one

4y = 18 Simplifying

y = 4 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.

Add the equations.

Divide each side by –1.

Simplify.

Multiply by –2.

Example 1 cont

Now substitute 9 for x in either equation to find the value of y.

Answer: The solution is (9, 5).

First equation

Simplify.

Subtract 18 from each side.

Simplify.

Example 2

Use elimination to solve the system of equations.

Method 1 Eliminate x.

Multiply by 3.

Add the equations.

Divide each side by 29.Simplify.

Multiply by –4.

Example 2 cont

Answer: The solution is (–1, 4).

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.

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.

Answer: The solution is (–1, 4), which matchesthe result obtained with Method 1.

First equation

Simplify.Add 4 to each side.

Simplify.

Divide each side by 3.

Simplify.

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 thecoefficients for y are the same or additive inverses, you cannot use elimination using additionor subtraction.

•Since the coefficient of the x term in the firstequation is 1, you can use the substitutionmethod. You could also use the eliminationmethod using multiplication.

Example 3 contThe following solution uses substitution.

First equation

Subtract 5y from each side.

Simplify.

Combine like terms.

Distributive Property

Second equation

Subtract 12 from each side.

Simplify.

Example 3 cont

Simplify.

First equation

Subtract 5 from each side.

Simplify.

Answer: The solution is (–1, 1).

Divide each side by –22.

Simplify.

Simplify.

Example 4Transportation 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.

10Upstream

10Downstream

dtr

This system cannot easily be solved using substitution. It cannot be solved by just adding or subtracting the equations.

Example 4 contThe best way to solve this system is to use elimination using multiplication. Since the problem asks for b, eliminate c.

Multiply by .

Add the equations.

Multiply each side by

Simplify.

Answer: The rate of the boat is 17.5 mph.

Multiply by .

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