# The Substitution Method

A way to solve systems of linear equations in 2 variables

### The Substitution Method

First, let's review how the substitution property works in general

Review Example 1

Review Example 2

##### Substitution Example 1

Let's re-examine system pictured up above.

$\red{y} = 2x + 1 \text{ and } \red{y} = 4x -1$

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.

$$y = 2x + 1 \\ y = 2\cdot \red{1} + 1 = 2 + 1 =3 \\ \\ \boxed{ \text{ or you use the other equation}} \\ y = 4x -1 \\ y = 4\cdot \red{1}- 1 = 4 - 1 = 3 \\ \text{solution = }( 1,3)$$

You can also solve the system by graphing and see a picture of the solution below:

### Substitution Practice Problems

##### Problem 1

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

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

$$y = \red{x} -5 \\ y = \red{-2} -5 = -7$$

The solution is the point (-2, -7)

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

This system has an infinite number of solutions. because 12x +4 = 12x is always true for all values of x.

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

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

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

This system has an no solutions.

The solution of this system is the point of intersection : (3,8).

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