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Algebra

Learn how to solve a system of linear equations using the substitution method. Isolate one variable, substitute into the other equation, solve, and check your solution.

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Algebra

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  1. Algebra 7.2 Solving Systems Using Substitution

  2. You have already learned that the solution is the point of intersection of the two graphed lines. Solution to a System of Linear Equations • To solve: • Graph both equations • Identify intersection point (x,y) • Plug in to original equations to check

  3. There are two algebraic methods that allow you to solve a system easily without graphing. Today you will learn the method called SUBSTITUTION.

  4. Steps In one equation, isolate one variable. Substitute expression from Step 1 into second equation and solve for the other variable. Plug in value from Step 2 into revised equation from Step 1 and solve. Check solution in both original equations.

  5. Hint: It is usually easiest to isolate positive 1x or 1y. 3x + y = 5 2x – y = 10 Solve the linear system. y = 5 – 3x 2x – ( ) = 10 5 – 3x y = 5 – 3(3) 2x – 5 + 3x = 10 y = 5 - 9 5x – 5 = 10 y = - 4 5x = 15 The solution is (3, - 4) x = 3 Check: 3(3) + (-4) = 5 2(3) – (-4) = 10

  6. 2x + 6y = 15 x = 2y Solve the linear system. x = 2y 2y 2( ) + 6y = 15 x = 2( 3/2) 4y + 6y = 15 x = 3 10y = 15 y = 15/10 The solution is (3, 3/2) y = 3/2 Check: 2(3) + 6(3/2) = 15 3 = 2(3/2)

  7. x + 2y = 4 -x + y = -7 You try! y = x – 7 x - 7 x + 2( ) = 4 y = 6 – 7 x + 2x - 14 = 4 y = -1 3x – 14 = 4 3x = 18 The solution is (6, - 1) x = 6 Check: 6 + 2(-1) = 4 -6 + (-1) = -7

  8. Mixture Problems Systems are often used to solved mixture problems. These are problems when you mix two quantities. You know the total quantity and the total value, but not how much of each type. To solve: Write one equation to describeQUANTITY. Write other equation to describe VALUE.

  9. Set up a system and solve the mixture problem. An audio store sells two styles of I-pod Nanos. The 2 GB model costs $150 and the 4GB model costs $225. Last Saturday the store sold 22 Nanos for a total of $3900. How many of each model did they sell? Let x be the # of 2 GB models sold Let y be the # of 4 GB models sold Quantity: x + y = 22 Value: 150x + 225y = 3900 There were 14 2GB Nanos and 8 4 GB Nanos sold. Now, you solve.

  10. Homework pg. 408 #17 – 35 odd, 42

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