Algebra Adventure: Variables on Both Sides!

Hi Phoebe! Ready for your next algebra adventure? Today, we'll learn how to solve equations that have variables hanging out on BOTH sides of the equals sign. It builds right on top of the 1-step and 2-step equations you already know!

Materials You'll Need:

• Whiteboard or paper
• Markers or pens/pencils

Quick Review (5 mins)

Remember how we solved equations like 3x + 5 = 17? We wanted to get 'x' by itself.

1. First, we undo the addition: Subtract 5 from both sides: 3x + 5 - 5 = 17 - 5 --> 3x = 12
2. Then, we undo the multiplication: Divide both sides by 3: 3x / 3 = 12 / 3 --> x = 4

Great job remembering!

New Challenge: Variables on Both Sides! (10 mins)

Now, what if we see something like this: 5x + 2 = 3x + 10?

See? We have 'x' terms on the left (5x) AND the right (3x). Our goal is still the same: get 'x' by itself. But first, we need to get all the 'x' terms together on one side and all the constant numbers together on the other side.

Think of it like a balancing scale. To keep it balanced, whatever we do to one side, we MUST do the exact same thing to the other side.

Our Strategy:

1. Move all the variable terms (like 5x and 3x) to one side (usually the left).
2. Move all the constant terms (like 2 and 10) to the other side (usually the right).
3. Solve the resulting simpler equation (it will look like a 1-step or 2-step equation!).

Let's Solve One Together (10 mins)

Example: 5x + 2 = 3x + 10

Step 1: Get variable terms together. We have 5x on the left and 3x on the right. Let's move the 3x. To get rid of +3x on the right, we subtract 3x from BOTH sides:

5x + 2 - 3x = 3x + 10 - 3x

Simplify both sides: (5x - 3x) + 2 = (3x - 3x) + 10

2x + 2 = 10

Look! It's a 2-step equation now!

Step 2: Get constant terms together. We have +2 on the left. Subtract 2 from BOTH sides:

2x + 2 - 2 = 10 - 2

Simplify: 2x = 8

Step 3: Solve for x. Divide BOTH sides by 2:

2x / 2 = 8 / 2

x = 4

Step 4: Check our answer (Super important!) Plug x = 4 back into the ORIGINAL equation: 5x + 2 = 3x + 10

5(4) + 2 = 3(4) + 10

20 + 2 = 12 + 10

22 = 22

It works! So, x = 4 is the correct solution.

Your Turn (Guided Practice - 10 mins)

Let's try this one: 7y - 4 = 2y + 11

What do you think we should do first to get the 'y' terms together? (Pause for answer - Hint: Subtract 2y from both sides)

Okay, let's do it: 7y - 4 - 2y = 2y + 11 - 2y --> 5y - 4 = 11

Now what? How do we get the constant numbers together? (Pause for answer - Hint: Add 4 to both sides)

Right! 5y - 4 + 4 = 11 + 4 --> 5y = 15

Last step? (Pause for answer - Hint: Divide by 5)

Perfect! 5y / 5 = 15 / 5 --> y = 3

Can you check if y = 3 makes the original equation true?

7(3) - 4 = 2(3) + 11 --> 21 - 4 = 6 + 11 --> 17 = 17. Yes!

Practice Power! (5 mins)

Try solving these on your paper. Remember the steps!

1. 4a + 3 = a + 15
2. 6b - 2 = 10 + 4b

We can check them together when you're done.

(Answers: 1. a=4, 2. b=6)

Wrap Up

Awesome job today, Phoebe! You learned how to solve equations with variables on both sides. The key is to collect the variable terms on one side and the constants on the other, turning it into a simpler equation you already know how to solve. Keep practicing, and you'll be an algebra pro in no time!