Solving Systems using Substitution

Standard A.REI.6: Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables.

Strategy

Pick an equation and solve for one of the variables.
Substitute into the other equation.
Solve for the single variable in this new equation.
Once you have x or y, plug it in to any equation to find the other variable!

Makes sense? Let's go over it in depth!

Example 1 (one solution)

Step 1: Solve for a variable

Notice that y is already by itself in the first equation. We don't need to do work!

Step 2: Substitute

Since we know what y is equal to in the first equation, we can substitute, or replace, that into the other equation.

Step 3: Solve for the single variable

Notice how the second equation now has only x as its variable. When there is only one variable, we can solve for it, or isolate it. Here's all the steps:

Step 4: Plug in the value

This step means to plug in the x variable we just found.
Plug this into any equation, but it's best to choose the easiest one. So in this example, we're choosing the first equation, y = 6x + 7.
Once we have solved for y, we have our solution!

Our solution:

Example 2 (one solution)

Step 1: Solve for a variable

Always choose the easiest choice. In the last example we solved for y. But solving for x in the second equation looks easier, so let's choose that.

Step 2: Substitute

Remember, you're always going to substitute into the other equation!

Step 3: Solve for single variable

Here's the steps for how we solved:

Distribution property for 3(3-2y)
Combine like terms -6y + 2y
Subtract 9 on both sides
Divide by -4 on both sides

Step 4: Plug in the value

We know what y is equal to, so let's plug that into either equation to find x. Again, choose the easiest looking equation, but either one is okay to use!

Equation #2 looks like the best one again.

Our solution:

Example 3 (infinite solutions)

Step 1: Solve for a variable

Step 2: Substitute

and

Step 3: Solve for single variable

When you see a solution that looks like this, it means they're the same line!

So when you see both sides of the equation are equal to each other, exactly equal, then it's the same line!

Tips for immediately seeing that they're the same equation:

Line up the terms

Example:

2x + y = 3
y = -2x + 3

We can subtract 2x on both sides of the equation of the first equation to get:

-2x + 2x + y = -2x + 3
0x + y = -2x + 3
y = -2x + 3

So they're the same equation.

Find a scaling factor

In the first example we had:

4x + 8y = 12
x + 2y = 3

The equations already have terms lined up but they have different numbers. However, there's a scaling factor (a number that you multiply or divide something by).

If you multiplied the second equation with a scaling factor of 4, we get: