Transcript Lesson 4
Lesson 7-4 Elimination Using Multiplication Click the mouse button or press the 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 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