1. Solve Linear Systems by Substitution

2. What does solving a system mean?
In yesterday’s homework, 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.”
A “System” is a pair of linear equations.

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.
We will learn this method today.
Elimination
This method is used when the equation is in standard form.
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. Practice #1:
Step 3: Substitute “7” in for “x” in the first equation, then solve.
y = x – 3
y = 7 – 3
y = 4
The solution to this linear system is an ordered pair, (7, 4). This is where the two lines intersect.
Step 4: Check the solution in each equation.
4 = 7 – 3
4 = 4 (true)
4x + y = 32
4(7) + 4 = 32
= 32
32 = 32 (true)
Find the solution to the linear system:
y = x – 3 4x + y = 32
Step 1: One of the equations is already solved for one variable, y = x – 3
Step 2: Substitute “x – 3” in for “y” in the 2nd equation.
4x + (x – 3) = 32
4x + x – 3 = 32
5x – 3 = 32
5x = 35
5 5
x = 7

6. 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

7. 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.
x – y = 2
+ y +y
x = 2 + y
Use distributive property.

8. 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)