Substitution Method

September 9, 2014Page 14-15 in Notes

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?

Solving Systems Using Substitution

• Title of Notes – pg. 15

Essential Question

How do I solve systems of linear equations using the substitution method?

System of Linear Equations

• Definition: a set of two or more equations with the same variables

• Example:2x + y = 55x – 3y = 8

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.

Substitution Steps

1. Isolate the “easiest” variable in either equation.

2. Substitute that variable in the other equation and solve for the remaining variable.

3. Substitute this value into the starting equation and solve for your first variable to find the rest of your ordered pair.

4. Check your point in both original equations.

Example 1: Solve the system of equations by substitution.

x – 2y = 5 solution: _______4x + 3y = 9

1 x = 2y + 5 (Step 1)

4(2y + 5) + 3y = 9 (Step 2) 8y + 20 + 3y = 9 11y + 20 = 9

11y = -11y = -1

x – 2(-1) = 5 (Step 3) x + 2 = 5

x = 3So, the solution to the system is (3, -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

(3, -1)

3x + y = 19 solution: _______3x – 2y = 25

Example 2: Solve the system of equations by substitution.

y = -3x + 19 (Step 1)

3x – 2(-3x+19) = 25 (Step 2) 3x + 6x – 38 = 25 9x – 38 = 25

9x = 63x = 7

3(7) + y = 19 (Step 3) 21 + y = 19

y = -2So, the solution to the system is (7, -2).

Check: (Step 4) 3x + y = 19 3x – 2y = 253(7) + (-2) = 19 3(7) – 2(-2) = 25 21 – 2 = 19 21 + 4 = 25

19 = 19 25 = 25

(7, -2)

Example 3: Solve the system of equations by substitution.

3x = y – 4 (Step 1) 3x + 4 = y

2x + 2y = 8 solution: _______3x – y = -4

2x + 2(3x + 4) = 8 (Step 2) 2x + 6x + 8 = 8 8x + 8 = 8

8x = 0x = 0

3(0) – y = -4 (Step 3) -y = -4

y = 4So, the solution to the system is (0, 4).

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

(0, 4)

Practice: On Your Own Paper

1. y = 2x + 15 y = x + 1

2. y = 6 x + 6y = 12

3. x = -6 2x – 3y = 7

4. x – y = 2 4x – 3y = 8

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

Reflection

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