7A Solving Equations Study Guide

Key Vocabulary

Equation — a math sentence that shows two expressions are equal, using an = sign. (ecuación)

Variable — a letter that stands for an unknown number. (variable)

Solve — to find the value of the variable that makes the equation true. (resolver)

Inverse Operation — the opposite operation that undoes another operation. (operación inversa)

Isolate — to get the variable alone on one side of the equation. (aislar)

Balance — both sides of the equation must stay equal, like a scale. (equilibrio)

Solution — the value that makes the equation true. (solución)

Visual Example — Balance Scale

Think of an equation like a balance scale. Both sides must weigh the same!

        x + 5  =  12
       -------   ------
      |  ???  | |  12  |
       -------   ------
    ============||============
                /\

  To find x, remove 5 from BOTH sides:
        x + 5 - 5  =  12 - 5
              x     =   7

Check: Plug 7 back in → 7 + 5 = 12 ✓ It works!

Step-by-Step: How to Solve an Equation

1 Look at the operation — What is happening to the variable? (adding, subtracting, multiplying, dividing)
2 Use the inverse operation — Do the opposite to both sides to isolate the variable.
3 Simplify — Calculate to find the value of the variable.
4 Check your answer — Substitute the solution back into the original equation to make sure it works.

Example: Solve n − 8 = 23
Step 1: The variable has − 8 (subtraction).
Step 2: Add 8 to both sides → n − 8 + 8 = 23 + 8
Step 3: n = 31
Step 4: Check → 31 − 8 = 23 ✓

Formula Box — Inverse Operation Pairs

Addition ↔ Subtraction (opposites)

Multiplication ↔ Division (opposites)

If the equation has...
x + 5 = 12 → Subtract 5 from both sides
x − 3 = 10 → Add 3 to both sides
4x = 20 → Divide both sides by 4
x ÷ 6 = 3 → Multiply both sides by 6

Watch Out!

Whatever you do to one side, you MUST do to the other side. The equation is a balance — keep it equal!
Always check your answer by substituting it back into the original equation.
"Solve" does not mean "compute." You must find the value that makes the equation true.
Read carefully: 4m means 4 × m, and y/5 means y ÷ 5.

Practice Problems

1. Solve: x + 15 = 42. What is x?

2. Solve: n − 8 = 23. What is n?

3. Solve: 4m = 36. What is m?