Let's re-examine system pictured on the left.
$
\color{red}{y} = 2x + 1 \text{ and } \color{red}{y} = 4x -1
$
Step 1) We are going to use substitution like we did in review example 2 above

Now we have 1 equation and 1 unknown, we can solve this problem as the work below shows.

The last step is to again use substitution, in this case we know that x = 1 , but in order to find the y value of the solution, we just substitute x =1 into either equation.
$$
y = 2x + 1
\\
y = 2\cdot \color{red}{1} + 1 = 2 + 1 =3
\\
\text{ or you can substitute x =1 into the other equation}
\\
y = 4x -1
\\
y = 4\cdot \color{red}{1}- 1 = 4 - 1 = 3
\\
\text{solution = }( 1,3)
$$

Substitution Example 2

What is the solution of the system of equations:
$
y = 2x + 1
\\
2y = 3x - 2
$

Step 1) Identify the best equation for substitution and then substitute into other equation.

Step 2) Solve for x

Step 3) Substitute the value of x (-4 in this case) into either equation

$$
y = 2x + 1
\\
y = 2\cdot \color{Red}{-4} + 1 = -8 + 1 = -7
\\
2y = 3x - 2\\
2y = 3\cdot-4 -2
\\
\text{or you can use the other equation}
\\
2y = -12 -2
\\
2y = -14
\frac{1}{2}\cdot2y =\frac{1}{2}\cdot-14
\\
y = -7
$$

Solve the system below using substitution
$$
y = x+1
\\
y = 2x +2
$$

Answer

The solution of this system is the point of intersection : (-3,-4).

Practice Problems

Problem 1)

Use substitution to solve the following system of linear equations:

Line 1: y=3x – 1

Line 2: y= x – 5

Step 1

Set the Two Equations Equal to each other then solve for x

Step 2

Substitute the x value, -2, into the value for 'x' for either equation to determine y coordinate of solution
$$
y = \color{Red}{x} -5
\\
y = \color{Red}{-2} -5 = -7
$$
The solution is the point (-2, -7)

problem 2)
Use the substitution method to solve the system:

Line 1: y = 5x – 1

Line 2: 2y= 3x + 12

Answer

This system of lines has a solution at the point (2, 9).

These lines have the same slope (slope =1) so they never intersect.

Problem 5)
Use the substitution method to solve the system:

Line 1: y= x + 1

Line 2: 2y= 3x

Answer

The solution of this system is (1,3).

Problem 6)
Use substitution to solve the system:

Line 1: y = 3x + 1

Line 2: 4y = 12x + 3

Answer

Whenever you arrive at a contradiction such as 3 = 4, your system of linear equations has no solutions.
When you use these methods (substitution, graphing , or elimination) to find the solution what you're really asking is at what