**Video** on Solving by Substitution

### 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.

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

Step 1We 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.

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

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

### Substitution **Practice** Problems

##### Practice 1

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

**Practice** Problems

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

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)**

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.

This system of linear equation has no solution.

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

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