# Substitution Method

How to solve a system of linear equations and inequalities by using their graphs: 3 examples and their solutions.

## Example 1

### Solution

Choose one of the equation.
Let's choose x - y = 4.

Change it to [x = ...] form.
Then x = y + 4.

You can also change it to [y = ...] form.

Put x = y + 4
into the other equation 2x + y = 5.

Then 2(y + 4) + y = 5.

Solve the equation 2(y + 4) + y = 5.
Then y = -1.

Put this y = -1
into x = y + 4.

Then x = (-1) + 4.

-1 + 4 = 3
So x = 3.

y = -1
x = 3

So [x = 3, y = -1] is the answer.

## Example 2

### Solution

Choose one of the equation.
Let's choose x - y = 4.

Change it to [x = ...] form.
Then x = y + 4.

Put x = y + 4
into the other equation 2x - 2y = 8.

Then 2(y + 4) - 2y = 8.

Solve the equation 2(y + 4) - 2y = 8.
Then the variable is removed
and you get 0 = 0.

0 = 0 is always true.

Just like this case,
when you get an equation that is always true,
then the system has infinitely many solutions.

So [infinitely many solutions] is the answer.

## Example 3

### Solution

Choose one of the equation.
Let's choose x - y = 4.

Change it to [x = ...] form.
Then x = y + 4.

Put x = y + 4
into the other equation x - y = -3.

Then (y + 4) - y = -3.

Solve the equation (y + 4) - y = -3.
Then the variable is removed
and you get 4 = -3.

4 = -3 is always false.

Just like this case,
when you get an equation that is always false,
then the system has no solution.

So [no solution] is the answer.