Algebra Detectives: The Case of the Balancing Variables!

Algebra Detectives: The Case of the Balancing Variables!

Hi Phoebe! You've become a pro at solving equations where the variable (our mystery number, 'x') is hiding on just one side. But sometimes, 'x' tries to be sneaky and appears on BOTH sides of the equals sign! Today, we're going to learn how to handle these tricky situations. Think of it like balancing a scale – we need to keep both sides equal while we figure out the mystery.

Let's Review!

Remember how we solve equations like 3x + 5 = 14? We want to get 'x' by itself. We use inverse operations:

1. Subtract 5 from both sides: 3x = 9
2. Divide by 3 on both sides: x = 3

The big rule is: Whatever you do to one side of the equation, you MUST do to the other side to keep it balanced!

The New Challenge: Variables on Both Sides!

What if we have an equation like: 5x + 2 = 3x + 10?

Oh no! 'x' is having a party on both sides! Our goal is still the same: get ONE 'x' all by itself. Here's how we can do it:

Step 1: Corral the Variables! Decide which side you want the variables on (it doesn't matter which, but picking the side that keeps the variable term positive is often easier). Let's move the '3x' term from the right side to the left side. Since it's a positive 3x, we do the inverse: subtract 3x from BOTH sides.

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

Look! Now it looks like a familiar two-step equation!

Step 2: Isolate the Variable Term! Now we need to get the '2x' term alone. Move the constant term (+2) away by doing the inverse: subtract 2 from BOTH sides.

2x + 2 = 10
- 2 = -2
-----------
2x = 8

Step 3: Solve for the Variable! Finally, get 'x' completely by itself. Since 'x' is multiplied by 2, we do the inverse: divide BOTH sides by 2.

2x = 8
-- --
2 2

x = 4

Step 4: Check Your Answer! (Super Important Detective Work!) Substitute x = 4 back into the ORIGINAL equation to make sure it's true.

Original: 5x + 2 = 3x + 10
Check: 5(4) + 2 = 3(4) + 10
20 + 2 = 12 + 10
22 = 22 (It works! Our solution is correct!)

Let's Try Another Together!

Equation: 7 + 2y = 4y - 3

1. Move Variables: Let's move the '2y' to the right side this time by subtracting 2y from both sides.
   7 + 2y = 4y - 3
   - 2y -2y
   -------------
   7 = 2y - 3
2. Move Constants: Now let's move the '-3' to the left side by adding 3 to both sides.
   7 = 2y - 3
   +3 +3
   ----------
   10 = 2y
3. Solve: Divide both sides by 2.
   10 = 2y
   -- --
   2 2
   
   5 = y (Or y = 5, same thing!)
4. Check:
   Original: 7 + 2y = 4y - 3
   Check: 7 + 2(5) = 4(5) - 3
   7 + 10 = 20 - 3
   17 = 17 (Correct!)

Your Turn: Equation Practice!

Try solving these equations. Remember the steps and check your answers!

1. 6a + 3 = 4a + 11
2. 8 + 5k = 18 + 3k
3. 10 - 2z = 4z - 8
4. 9m - 4 = 2m + 17

(We can go over these together after you try them.)

Wrap Up

Great job today, Algebra Detective! You learned how to solve equations even when the variable tries to hide on both sides. The key steps are:

1. Move all variable terms to one side using inverse operations.
2. Move all constant terms to the other side using inverse operations.
3. Solve the remaining one-step equation.
4. Always check your answer!

Keep practicing, and soon these equations will seem easy!