Study Guide for 13-Year-Olds: Combining Like Terms & Solving Single- and Multi-Step Equations (Middle School Math)

Instructions

This study guide will walk you through the key concepts of expressions and equations. Read each section carefully, review the examples, and then try the practice problems to test your understanding. Good luck!

Section 1: Combining Like Terms

What are "Like Terms"?

Think of variables like types of fruit. You can add 3 apples and 5 apples to get 8 apples. But you can't add 3 apples and 2 oranges to get "5 apple-oranges." It doesn't work!

In math, like terms are terms that have the exact same variable(s) raised to the exact same power. The number in front, called the coefficient, can be different.

• Examples of Like Terms: 7x and -2x | y and 15y | -4a and 10a | 8 and -12 (these are called constants)
• Examples of UNLIKE Terms: 7x and 7y (different variables) | 5a and 8b (different variables) | 3x and 3x² (different powers)

How to Combine Like Terms

To combine like terms, you simply add or subtract their coefficients. The variable part stays the same!

Example: Simplify the expression 3x + 4 + 5x - 10

1. Identify the like terms.
   • The 'x' terms are 3x and 5x.
   • The constant terms are +4 and -10.
2. Combine each group.
   • 3x + 5x = 8x
   • 4 - 10 = -6
3. Write the new, simplified expression: 8x - 6

Practice: Simplifying Expressions

Simplify each expression by combining like terms.

1. 9a + 3a = ?
2. 10b - 4b + 2 = ?
3. -5x + 7y + 8x - 2y = ?
4. 6 + 4m - 3 - 9m = ?

Section 2: Solving Single-Step Equations

The Golden Rule of Equations

An equation is like a balanced scale. To keep it balanced, whatever you do to one side of the equation, you MUST do to the other side. The goal is to get the variable all by itself.

Undoing Operations

To get the variable alone, you use the "inverse" (or opposite) operation.

• To undo addition, you subtract.
  • Example: x + 5 = 15 => Subtract 5 from both sides => x = 10
• To undo subtraction, you add.
  • Example: y - 8 = 2 => Add 8 to both sides => y = 10
• To undo multiplication, you divide.
  • Example: 4a = 24 => Divide both sides by 4 => a = 6
• To undo division, you multiply.
  • Example: b / 3 = 7 => Multiply both sides by 3 => b = 21

Practice: Single-Step Equations

Solve for the variable in each equation.

5. c + 11 = 30
6. g - 9 = -5
7. -6k = 42
8. x / 5 = 10

Section 3: Solving Multi-Step Equations

The Strategy: Reverse Order of Operations

When you solve complex equations, you are "un-doing" the math. You work backward, following the order of operations in reverse. Think of it as SADMEP.

1. Simplify First! If one side of the equation has like terms, combine them before you start solving.
2. (S) Undo any Subtraction or (A) Addition.
3. (D) Undo any Division or (M) Multiplication.
4. (E) Deal with Exponents (you'll see this more later).
5. (P) Clear any Parentheses (you'll also see this more later).

Example: Solve the equation 4x - 7 = 13

1. There are no like terms to combine. So we start with SADMEP.
2. Undo Subtraction/Addition: The "- 7" needs to be undone. Add 7 to both sides.
   • 4x - 7 + 7 = 13 + 7
   • 4x = 20
3. Undo Multiplication/Division: The "4x" is multiplication. Divide both sides by 4.
   • 4x / 4 = 20 / 4
   • x = 5

Example with Simplifying First: Solve the equation 6x + 3 - 2x = 23

1. Simplify First! Combine the like terms 6x and -2x.
   • 4x + 3 = 23
2. Now, follow SADMEP on the new, simpler equation. Undo Addition by subtracting 3.
   • 4x + 3 - 3 = 23 - 3
   • 4x = 20
3. Undo Multiplication by dividing by 4.
   • 4x / 4 = 20 / 4
   • x = 5

Practice: Multi-Step Equations

Solve for the variable in each equation.

9. 5y + 4 = 29
10. -2a - 10 = 8
11. k / 3 + 1 = 7
12. 7x - 4x + 8 = 17

Answer Key

1. 12a
2. 6b + 2
3. 3x + 5y
4. -5m + 3
5. c = 19
6. g = 4
7. k = -7
8. x = 50
9. y = 5
10. a = -9
11. k = 18
12. x = 3