Elimination Using Multiplication 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. 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. 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. 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.

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: 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 391 14-38 even There is a graph