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Substitution Method

Substitution Method. September 9, 2014 Page 14-15 in Notes. Warm-Up (page 14). What is an equation? Which of the following equations is linear? A. 2x + y = 8 B. 2x 2 + 4x – 3 = 7 What is a linear equation?. Solving Systems Using Substitution. Title of Notes – pg. 15.

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Substitution Method

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  1. Substitution Method September 9, 2014 Page 14-15 in Notes

  2. Warm-Up (page 14) • What is an equation? • Which of the following equations is linear? • A. 2x + y = 8 • B. 2x2 + 4x – 3 = 7 • What is a linear equation?

  3. Solving Systems Using Substitution • Title of Notes – pg. 15

  4. Essential Question How do I solve systems of linear equations using the substitution method?

  5. System of Linear Equations • Definition: a set of two or more equations with the same variables • Example: 2x + y = 5 5x – 3y = 8

  6. Solving Systems of Equations • One method we use to solve systems of equations algebraically is called the substitution method. • The solution to a system of equations is the ordered pair (x, y) that makes both equations true. It is also the point on the graph where the two lines intersect.

  7. Substitution Steps • Isolate the “easiest” variable in either equation. • Substitute that variable in the other equation and solve for the remaining variable. • Substitute this value into the starting equation and solve for your first variable to find the rest of your ordered pair. • Check your point in both original equations.

  8. Example 1: Solve the system of equations by substitution. (3, -1) x – 2y = 5 solution: _______ 4x + 3y = 9 1 x = 2y + 5 (Step 1) Check: (Step 4) x – 2y = 5 4x + 3y = 9 (3) – 2(-1) = 5 4(3) + 3(-1) = 9 3 + 2 = 5 12 – 3 = 9 5 = 5 9 = 9 4(2y + 5) + 3y = 9 (Step 2) 8y + 20 + 3y = 9 11y + 20 = 9 11y = -11 y = -1 x – 2(-1) = 5 (Step 3) x + 2 = 5 x = 3 So, the solution to the system is (3, -1).

  9. Example 2: Solve the system of equations by substitution. (7, -2) 3x + y = 19 solution: _______ 3x – 2y = 25 y = -3x + 19 (Step 1) 3x – 2(-3x+19) = 25 (Step 2) 3x + 6x – 38 = 25 9x – 38 = 25 9x = 63 x = 7 3(7) + y = 19 (Step 3) 21 + y = 19 y = -2 So, the solution to the system is (7, -2). Check: (Step 4) 3x + y = 19 3x – 2y = 25 3(7) + (-2) = 19 3(7) – 2(-2) = 25 21 – 2 = 19 21 + 4 = 25 19 = 19 25 = 25

  10. Example 3: Solve the system of equations by substitution. (0, 4) 2x + 2y = 8 solution: _______ 3x – y = -4 3x = y – 4 (Step 1) 3x + 4 = y Check: (Step 4) 2x + 2y = 8 3x – y = -4 3(0) + 2(4) = 8 3(0) – (4) = -4 0 + 8 = 8 0 – 4 = -4 8 = 8 -4 = -4 2x + 2(3x + 4) = 8 (Step 2) 2x + 6x + 8 = 8 8x + 8 = 8 8x = 0 x = 0 3(0) – y = -4 (Step 3) -y = -4 y = 4 So, the solution to the system is (0, 4).

  11. Practice: On Your Own Paper 5. x + 2y = -9 3x + 2y = -7 6. 23x + 11y = 1 -2x – y = 0 7. 3x + y = -20 2x – 7y = 2 8. y + 3x = 9 4x + 2y = 17 • y = 2x + 15 y = x + 1 • y = 6 x + 6y = 12 • x = -6 2x – 3y = 7 4. x – y = 2 4x – 3y = 8

  12. Reflection • What did all the problems we looked at today have in common that made it easy to use the substitution method for solving?

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