6 Solve Systems of Equations: Elimination Sometimes we can multiply two sets of equations by a constant and add them together to eliminate a variableExample: Solve 2x + 4y = 20 and -3x + 8y = 262x + 4y = (equation one; 4y 2 = 8y) x + 8y = (equation one 2)x + 8y = (equation two) (-#) is a positive7x = Eliminate y by subtractingx = Divide both sides by 72(2) + 4y = Sub x= into equation one4y = Simplifyingy = Divide both sides by 4
7 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.
8 Example 1 contNow substitute 9 for x in either equation to find the value of y.First equationSimplify.Subtract 18 from each side.Simplify.Answer: The solution is (9, 5).
9 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.
10 Example 2 cont Now substitute 4 for y in either equation to find x. First equationSimplify.Subtract 12 from each side.Simplify.Divide each side by 4.Simplify.Answer: The solution is (–1, 4).
11 Example 2 – Another Way Method 2 Eliminate y. Multiply by 5. Add the equations.Divide each side by 29.Simplify.
12 Example 2 – Another Way cont Now substitute –1 for x in either equation.First equationSimplify.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.
13 Example 3Determine 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.
14 Example 3 cont The following solution uses substitution. First equationSubtract 5y from each side.Simplify.Second equationDistributive PropertyCombine like terms.Subtract 12 from each side.Simplify.
15 Example 3 cont Simplify. Divide each side by –22. Simplify. First equationSimplify.Subtract 5 from each side.Simplify.Answer: The solution is (–1, 1).
16 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.10UpstreamDownstreamdtrThis system cannot easily be solved using substitution. It cannot be solved by just adding or subtracting the equations.
17 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 .Multiply by .Add the equations.Multiply eachside bySimplify.Answer: The rate of the boat is 17.5 mph.
18 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 ormultiplication
19 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 eliminationsThree methods for solving systems of equations:GraphingSubstitutionElimination (using addition, subtraction or multiplication)Homework:Pg evenThere is a graph