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. 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 = 5
5x – 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.
1 x – 2y = 5
x = 2y + 5
(3, -1)
(Step 1) solution: _______
4x + 3y = 9
(Step 4)
4(2y + 5) + 3y = 9 (Step 2) Check:
x – 2y = 5
4x + 3y = 9
8y + 20 + 3y = 9
(3) – 2(-1) = 5
4(3) + 3(-1) = 9
11y + 20 = 9
3+2=5
12 – 3 = 9
11y = -11
5=5
9=9
y = -1
x – 2(-1) = 5
(Step 3)
x+2=5
x=3
So, the solution to the system is (3, -1).
Example 2: Solve the system of
equations by substitution.
(7, -2)
3x + y = 19 y = -3x + 19 (Step 1) solution: _______
3x – 2y = 25
3x – 2(-3x+19) = 25 (Step 2)
3x + 6x – 38 = 25
Check:
(Step 4)
9x – 38 = 25
3x + y = 19
3x – 2y = 25
9x = 63
3(7) + (-2) = 19 3(7) – 2(-2) = 25
x=7
21 – 2 = 19
21 + 4 = 25
3(7) + y = 19
(Step 3)
19 = 19
25 = 25
21 + y = 19
y = -2
So, the solution to the system is (7, -2).
Example 3: Solve the system of
equations by substitution.
2x + 2y = 8
3x – y = -4
(0, 4)
solution: _______
3x = y – 4 (Step 1)
3x + 4 = y Check:
(Step 4)
2x + 2(3x + 4) = 8 (Step 2)
2x + 2y = 8
3x – y = -4
2x + 6x + 8 = 8
3(0)
+
2(4)
=
8
3(0)
–
(4)
=
-4
8x + 8 = 8
0+8=8
0 – 4 = -4
8x = 0
8=8
-4 = -4
x=0
3(0) – y = -4
(Step 3)
-y = -4
y=4
So, the solution to the system is (0, 4).
Practice: On Your Own Paper
1. y = 2x + 15
y=x+1
5. x + 2y = -9
3x + 2y = -7
2. y = 6
6. 23x + 11y = 1
x + 6y = 12
3. x = -6
2x – 3y = 7
4. x – y = 2
4x – 3y = 8
-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?