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Transcript of Lesson 7-4 Elimination Using Multiplication. Transparency 4 Click the mouse button or press the...
Lesson 7-4
Elimination Using Multiplication
Click the mouse button or press the Click the mouse button or press the Space Bar to display the answers.Space Bar to display the answers.
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