# Substitution Method - Part 2

## Solve System of Equations Substitution Method - Part 2 - How it Works - Video

### Which Variable to Solve For

Which Variable to Solve For:

We have 4 options when dealing with 2 equations in a system of equations. We can solve for x or y for the nonlinear equation or we can solve for x or y for the linear equation. Each one has their own pros and cons, and that depends on you.

We are going to use x = 2y + 3.

### Example 1 - Substitute

Example 1 Substitute:

Here we have solved for the variable x in x - 2y = 3 and our result is x = 2y + 3. We have to substitute this equation into our other equation, x + y2 = 6.

x + y2 = 6

(2y - 3) + y2 = 6

Original

We substituted

### Example 1 - Solve for the 1st Variable

Example 1 Solve for the 1st Variable:

Here we have solved for the variable x in x - 2y = 3 and our result is x = 2y + 3. We have to substitute this equation into our other equation, x + y2 = 6.

x + y2 = 6

(2y - 3) + y2 = 6

y2 + 2y - 3 = 6

y2 + 2y - 9 = 0

(y + 3)(y - 1) = 0

Original

We substituted

Rearrange terms

Subtracted 9 to both terms

Factor

We have factored and now we have two equations y + 3 = 0 and y - 1 = 0 so y = -3 and y = 1.

### Example 1 - Plot 1st Equation

Example 1 Plot 1st Equation:

We have factored and now we have two equations y + 3 = 0 and y - 1 = 0 so y = -3 and y = 1.

Now that we know y = -3 and y =1. We can substitute in either question to find the x part of the ordered pair. Here we chose the linear equation.

x - 2y = 3

x - 2 * (-3) = 3

x + 6 = 3

x = -3

Substituted -3

Multiplied

x - 2y = 3

x - 2 * (1) = 3

x - 2 = 3

x = 5

Substituted 1

Multiplied

So our points are (-3, -3) and (5, 1).

### Example 1 - Plot 2nd Equation

Example 1 Plot 2nd Equation:

We know the points (-3, -3) and (5, 1) are also true for our 2nd equation. We know that is a parabola, which means we need our vertex. To find our vertex since we have a sideways parabola, we can use the formula y = -b / (2a). Which one is a? Which one is b? We look at the standard form, x = ay2 + by + c. We have x = -1y2 + 0y + 6, so a = -1, b = 0, and c = 6.

y = -b / [2a]

y = - (0) / [2 * (-1)]

y = 0

Substituted a and b

Multiplied

x = -y2 + 6

x = -(0)2 + 6

x = 6

Substituted a and b