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Mastering Linear Systems: Graphing, Substitution, and Elimination Techniques

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In this lesson, we will focus on solving linear systems using graphing, substitution, and elimination methods. We’ll tackle various example problems, particularly emphasizing the substitution method. Students will learn how to isolate variables and substitute them into equations to find the solution to systems of equations. We’ll work through practice problems collaboratively, including a word problem involving mowing and shoveling, and students will complete assignments from pages 439-440 for additional reinforcement.

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Mastering Linear Systems: Graphing, Substitution, and Elimination Techniques

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  1. Agenda forThursday 11/29/12 Target 7a: I can solve linear systems by graphing, substitution, and elimination. • Questions on solving using substitution • More on Solving Linear Systems Using Substitution • Assignment…

  2. Solving Linear Systems Using Substitution

  3. Solving Linear Systems Using Substitution x = 2y – 6 x – 2y = -6 4x + 6y = 4 Before you can use substitution, x or y needs to be alone in one of the equations. 4(2y– 6) + 6y = 4 8y – 24 + 6y = 4 14y – 24 = 4 +24 +24 14y = 28 14 14 y = 2 Plug 2y – 6 in for x. Solve for y. x = 2(2) – 6 x = -2 Plug 2 in for y. Solve for x. The solution is (-2,2)

  4. Lets try another one together! y = -5x – 12 -5x – y = 12 3x – 5y = 4 Get x or y alone in one of the equations. 3x – 5(-5x– 12) = 4 3x + 25x + 60 = 4 28x + 60 = 4 -60 -60 28x = -56 28 28 x = -2 y = -5(-2) – 12 y = -2 Plug -5x – 12 in for y. Solve for x. Plug -2 in for x. Solve for y. The solution is (-2,-2)

  5. Try these… x = y + 3 x – y = 3 x + 2y = -6 y = -3x – 7 3x + y = -7 -2x + 4y = 0 (y + 3) + 2y = -6 3y + 3 = -6 -3 -3 3y = -9 3 3 y = -3 x = -3 + 3 x = 0 (0,-3) -2x + 4(-3x – 7) = 0 -2x – 12x – 28 = 0 -14x – 28 = 0 +28 +28 -14x = 28 -14 -14 x = -2 y = -3(-2) – 7 y = -1 (-2,-1)

  6. A Word Problem! Last year, you mowed grass and shoveled snow for 12 households. You earned $225 for mowing a household’s lawn for an entire year and $200 for shoveling a household’s walk and driveway for an entire year. You earned a total of $2600 last year. How many households did you mow and shovel for last year? Step 1: Write a system of equations. x is mowing y is shoveling x + y = 12 225x + 200y = 2600 y = -x + 12 Step 2: Solve using substitution. 225x + 200(-x + 12) = 2600 225x – 200x + 2400 = 2600 25x + 2400 = 2600 25x = 200 x = 8 y = -8 + 12 y = 4 8 lawns mowed 4 walks & driveways shoveled

  7. Agenda forThursday 11/29/12 Target 7a: I can solve linear systems by graphing, substitution, and elimination. • Assignment… • Pg. 439-440

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