System of Equations: Substitution Method Lesson

Solving Systems of Equations

I can use the substitution method in order to solve systems of equations in order to help my team achieve a score of at least 80%.

A1.AREI.6* Solve systems of linear equations algebraically and graphically focusing on pairs of linear equations in two variables.

What is a system?

​There are many systems all around you!

Systems are made of up of parts that work together to accomplish some task. Your body is made up of systems! The school you're in is a part of a school system.

Each of these systems have parts that form its entirety. And guess what... Math has systems too!

Open Ended

Come up with two different systems in the world around us.

3x + 4y = 12

-x + 6y = -2

We'll be discussing at a system that contains at least two equations to describe it. These will look something like this:

​System of Equations

What are systems in math?

Multiple Choice

Choose the system of equations.

• 3x - 4y = 2

• y = 3x + 8

• 3x = 10

  2x + 4y = 9

• -14w

What is a solution?

If we're looking for a solution to the system, what we're looking for is an x-value and a y-value that will make both equations true.

Example:

3x + 2y = 5

-2x + y = 6

3(-1) + 2(4) = -3 + 8 = 5

-2(-1) + (4) = 2 + 4 = 6

​This system has a solution. If you plug in x = -1 & y = 4, you get two true equations:

Both equations are true for this x and y-value, so it is a solution. We can write this as a point: (-1, 4)

Multiple Choice

Which point would be a solution to the system:

5x - y = 12

3x + 2y = 2

• (3, 4)

• (2, -2)

• (4, 0)

• (-3, 6)

How do you find a solution?

There are several ways to find a solution to system. Today, we'll discuss using a method called "substitution." You've already used the core technique of substitution in this lesson, so I believe that you've got this!

With the substitution method, we'll do the following steps:

1) Pick a variable to solve for in either equation (use your noodle!)

2) Isolate that variable.

3) Substitute this into the other equation.

4) Solve your new equation for the variable.

5) Plug this answer back into an original equation to find the solution.

​Wait, you're stressed out? Maybe a little confused? No sweat! You got this! Let's break it down step by step:

Step 1) Pick a variable to solve for in either equation (use your noodle!)

We want to get a variable by itself in one equation. It doesn't matter which equation, and it doesn't matter which variable. Let's look at an example:

​3x + 2y = 5

-2x + y = 6

​Which variable looks like it would be the easiest to get by itself?

Poll

Using the problem from before, which variable do you think is easiest to solve

for? Another way to say that is, which variable would be easiest to get by itself?

3x + 2y = 5

-2x + y = 6

• 3x in 1st equation

• 2y in 1st equation

• -2x in 2nd equation

• y in 2nd equation

If you said the y in the 2nd equation, me too! It's just a regular y. Let's move to the next step:

Step 2) Isolate that variable.

Isolate means "get by itself on one side." All we would need to do is move the x's to the other side of the equation:

-2x + y = 6

+2x +2x

y = 2x + 6

BOOM! y is by itself! That's step 2. Give it a try...

Fantastic! Now, what do we do with this??

Step 3) Substitute this into the other equation.

A system has two equations, and you just worked with one. Since we now know what one variable is equal to, we're going to substitute that into the other equation.

Looking back at our example:

​3x + 2y = 5

-2x + y = 6

​We solved the bottom equation to get:

y = 2x + 6

​Look at the substitution step on the next slide:

​3x + 2y = 5

y = 2x + 6

​Substitute into the other

equation for the y

​This would turn into:

​3x + 2(2x + 6) = 5

​** Remember: when we plug in, ALWAYS USE PARENTHESIS!!

Multiple Choice

Let x = 3y - 4

and

5x - 4y = 2

Select the answer that has correctly substituted (plugged in) the first equation into the second.

• (3y - 4) - 4y = 2

• 5x(3y - 4) -4y = 2

• (3y - 4) = 2

• 5(3y - 4) - 4y = 2

Wonderful! So, we've substituted one equation into the other one. What comes next?

Step 4) Solve your new equation for the variable.

We now have a new equation with just one variable in it! You can solve these all day long:

3x + 2(2x + 6) = 5

3x + 4x + 12 = 5

7x + 12 = 5

7x = -7

x = -1

ONE STEP TO GO!!!

You're almost there! You now have half of your solution, because the solution has both an x-value AND a y-value. How do we get the other half?

Step 5) Plug this answer back into an original equation to find the solution.

Let's look at our example and what we've solved for:

​3x + 2y = 5

-2x + y = 6

​We now know x = -1. Let's plug this into one of our equations and solve for y:

3(-1) + 2y = 5

-3 + 2y = 5

2y = 8

y = 4

​So, our solution is:

x = -1 & y = 4, or

(-1, 4)

​That's not too bad, right? Let's review those steps one more time:

Step 1) Pick a variable to solve for in either equation (use your noodle!)

**It doesn't matter which equation or which variable, you choose!

Step 2) Isolate that variable.

**Get it by itself on one side of the equation.

Step 3) Substitute this into the other equation.

**Plug it into it's variable USING PARENTHESIS! You should only have 1 variable in this equation.

Step 3) Solve your new equation for the variable.

**Follow the normal solving steps.

Step 5) Plug this answer back into an original equation to find the solution.

**Solutions have two parts: x & y. Don't forget the other half!

Ready to give it a try on your own? Let's take it step by step, just in case you get stuck...

Multiple Choice

Step 1: Pick a variable to solve for in either equation.

Copy down the system:

2x - 5y = 1

x + 3y = 6

Which variable is the easiest to solve for?

• 2x in the 1st equation

• -5y in the 1st equation

• x in the 2nd equation

• 3y in the 2nd equation

​The easiest variable to solve for is the x in the second equation. Let's move to step 2...

Multiple Choice

Step 2: Isolate the variable.

x + 3y = 6

Which equation has correctly isolated the x?

• 3y = -x + 6

• x = 3y + 6

• x = 3y - 6

• x = -3y + 6

​You would need to subtract the 3y:

x + 3y = 6

-3y -3y

x = -3y + 6

Now on to Step 3...

Multiple Choice

Step 3: Substitute this into the other equation.

Which equation below shows correctly substituting in the previous step's answer into the other equation?

• 2(-3y + 6) - 5y = 1

• (-3y + 6) - 5y = 1

• 2(-3y + 6) = 1

• 2x - 5(-3y + 6) = 1

​When you substitute into the other equation, use parenthesis:

​x = -3y + 6

​2x - 5y = 1

​2(-3y + 6) - 5y = 1

​becomes

​Check your solving steps:

2(-3y + 6) - 5y = 1

-6y + 12 - 5y = 1

-11 y + 12 = 1

-11y = -11

y = 1

​Almost done! Last step....

​You did it!! I'm so proud of you! Now, do you think you can do one all on your own? I bet you can! Give it a try...

Way to go!

​You now know how to use the substitution method!

We'll continue tomorrow by learning how to solve a system of equations using a different method. The top 2 groups on today's assignment will receive a prize tomorrow! I hope you worked hard!