1. Solve Linear Systems by Substitution

2. What does solving a system mean?
A “System” is a pair of linear equations.
Previously we investigated Cool Copy Company and Bountiful Brochures.
What happened when you graphed the lines?
The lines intersected.
What does that intersection mean?
The two companies sold the same number of copies for the same amount of money.
The point of intersection is called “Solving the System.”

3. 3 ways to solve a system: Graphing Substitution Elimination
This method is only successful if you draw an accurate graph.
Substitution
This method has you substitute one equation into another equation and solve algebraically.
This method is more precise
We will learn this method today.
Elimination
We will learn this method later in the unit.

4. Steps to solve a linear system by substitution
Step 1: Solve one of the equations for one of its variables.
Step 2: Substitute this expression into the other equation and solve for the other variable.
Step 3: Substitute this value into the revised first equation and solve.
Step 4: Check the solution pair in each of the original equations.
Remember, solving a system of linear equations means that you are finding where the two lines intersect.

5. Find the solution to the linear system:
Practice #1:
Find the solution to the linear system:
y = x – 3 4x + y = 32
Step 1: Solve one equation for one variable (get one in a y= or x= form)
Step 2: Substitute “x – 3” in for “y” in the 2nd equation, then Solve that equation for x
Step 3: Substitute “7” in for “x” in the first equation, then solve for the y.
One of the equations is already solved for one variable, no work to do
y = x – 3 4x + y = 32
4x + (x – 3) = 32
4x + x – 3 = 32
5x – 3 = 32
5x = 35
5 5
x = 7
y = x – 3
y = (7) – 3
y = 4
The solution to the system is an ordered pair (7, 4)

6. PRACTICE #1 CONTINUED
The solution to this linear system is an ordered pair, (7, 4).
This is where the two lines intersect when graphed
Step 4: Check the solution in each equation. Plug 7 in as x and plug 4 in as y, and it should work for both equations
y = x – 3
4 = 7 – 3
4 = 4 (true)
4x + y = 32
4(7) + 4 = 32
= 32
32 = 32 (true)

7. Practice #2:
Step 3: Substitute “3” in for “x” in the first equation, then solve.
y = x + 4
y = 3 + 4
y = 7
The solution to this linear system is an ordered pair, (3, 7). This is where the two lines intersect.
Step 4: Check the solution in each equation.
7 = 3 + 4
7 = 7 (true)
3x + y = 16
3(3) + 7 = 16
9 + 7 = 16
16 = 16 (true)
Find the solution to the linear system:
y = x x + y = 16
Step 1: One of the equations is already solved for one variable,
y = x + 4
Step 2: Substitute “x + 4” in for “y” in the 2nd equation.
3x + (x + 4) = 16
3x + x + 4= 16
4x + 4 = 16
4x = 12
4 4
x = 3

8. Use distributive property.
Practice #3:
Step 2: Substitute “2 + y” in for “x” in the 2nd equation.
2x + y = 1
2(2 + y) + y = 1
4 + 2y + y = 1
4 + 3y = 1
3y = -3
y = -1
Find the solution to the linear system:
x – y = 2 2x + y = 1
Step 1: Solve the 1st equation for one variable. (get the x= all alone on one side)
x – y = 2
+ y +y
x = 2 + y
Use distributive property.

9. Step 3: Substitute “-1” in for “y” in the first equation, then solve.
x – y = 2
x – –1 = 2
x + 1 = 2
x = 1
The solution to this linear system is an ordered pair, (1, -1). This is where the two lines intersect.
Step 4: Check the solution in each equation.
x – y = 2
1 – –1 = 2
1 + 1 = 2
2 = 2 (true)
2x + y = 1
2(1) + -1 = 1
= 1
1 = 1 (true)