Instructions
This worksheet focuses on solving two-step linear equations involving a single variable. Remember that to isolate the variable, you must use inverse operations in the correct order:
- Undo Addition or Subtraction (Inverse of the constant term).
- Undo Multiplication or Division (Inverse of the coefficient).
Follow the steps for each section, showing your work clearly.
Section 1: The Equation Toolkit (Guided Practice)
Review the example below, which shows the proper steps for solving a two-step equation. Then, solve the remaining problems using the same structure.
Key Tool: Inverse Operations
| Operation in Equation | Inverse Operation (Step 1 or 2) |
|---|---|
| Addition (+) | Subtraction (-) |
| Subtraction (-) | Addition (+) |
| Multiplication (e.g., 3x) | Division (÷) |
| Division (e.g., x/4) | Multiplication (×) |
Guided Solutions
| Equation | Step 1: Undo +/- | Step 2: Undo x/÷ | Solution (x = ?) |
|---|---|---|---|
| Example: $4x + 6 = 30$ | Subtract 6: $4x = 24$ | Divide by 4: $x = 6$ | $x = 6$ |
| 1. $2y - 10 = 4$ | |||
| 2. $5n + 3 = 18$ | |||
| 3. $\frac{k}{3} - 2 = 1$ | |||
| 4. $15 = 3m + 9$ |
Section 2: Skill Builder (Intermediate Equations)
Solve the following equations. These problems may involve negative integers or subtraction of the variable term. Show your steps below the equation.
- $7 + 3x = 19$
$x = $
- $-2a - 5 = 15$
$a = $
- $11 = 25 - 4p$
$p = $
- $\frac{w}{-5} + 8 = 10$
$w = $
- $6b - 18 = 0$
$b = $
Section 3: Real-World Mystery (Application)
For each scenario, first write a two-step equation to represent the situation, and then solve the equation to find the unknown value.
- The Bowling Alley Fee: A bowling alley charges a one-time shoe rental fee of $3, plus $4 per game. If your total bill was $19, how many games ($g$) did you play?
Equation:
Solution ($g$):
- The Concert Ticket Group Rate: A group purchased concert tickets. They paid a flat $10 service fee for the entire order, plus $25 for each ticket ($t$). If the total cost was $160, how many tickets did they buy?
Equation:
Solution ($t$):
- Savings Goal: Sarah already has $50 saved. She decides to save $15 every week ($w$) from her allowance. She needs to save a total of $200 for a new video game console. How many weeks will it take?
Equation:
Solution ($w$):
Section 4: The Equation Challenge (Advanced)
These problems require careful handling of decimals or fractions. Solve for the variable.
- $0.5x + 7 = 13$
$x = $
- $\frac{2}{3} z - 4 = 12$
$z = $
Answer Key
Section 1: The Equation Toolkit (Guided Practice)
| Equation | Step 1: Undo +/- | Step 2: Undo x/÷ | Solution (x = ?) |
|---|---|---|---|
| Example: $4x + 6 = 30$ | Subtract 6: $4x = 24$ | Divide by 4: $x = 6$ | $x = 6$ |
| 1. $2y - 10 = 4$ | Add 10: $2y = 14$ | Divide by 2: $y = 7$ | $y = 7$ |
| 2. $5n + 3 = 18$ | Subtract 3: $5n = 15$ | Divide by 5: $n = 3$ | $n = 3$ |
| 3. $\frac{k}{3} - 2 = 1$ | Add 2: $\frac{k}{3} = 3$ | Multiply by 3: $k = 9$ | $k = 9$ |
| 4. $15 = 3m + 9$ | Subtract 9: $6 = 3m$ | Divide by 3: $m = 2$ | $m = 2$ |
Section 2: Skill Builder (Intermediate Equations)
-
$7 + 3x = 19$ (Subtract 7, $3x=12$; Divide by 3) $x = 4$
-
$-2a - 5 = 15$ (Add 5, $-2a=20$; Divide by -2) $a = -10$
-
$11 = 25 - 4p$ (Subtract 25, $-14 = -4p$; Divide by -4) $p = 3.5$ (or $7/2$)
-
$\frac{w}{-5} + 8 = 10$ (Subtract 8, $\frac{w}{-5} = 2$; Multiply by -5) $w = -10$
-
$6b - 18 = 0$ (Add 18, $6b=18$; Divide by 6) $b = 3$
Section 3: Real-World Mystery (Application)
-
Equation: $4g + 3 = 19$ (Subtract 3: $4g = 16$; Divide by 4) Solution ($g$): $g = 4$ games
-
Equation: $25t + 10 = 160$ (Subtract 10: $25t = 150$; Divide by 25) Solution ($t$): $t = 6$ tickets
-
Equation: $15w + 50 = 200$ (Subtract 50: $15w = 150$; Divide by 15) Solution ($w$): $w = 10$ weeks
Section 4: The Equation Challenge (Advanced)
-
$0.5x + 7 = 13$ (Subtract 7: $0.5x = 6$; Divide by 0.5 or Multiply by 2) $x = 12$
-
$\frac{2}{3} z - 4 = 12$ (Add 4: $\frac{2}{3} z = 16$; Multiply by the reciprocal $\frac{3}{2}$) $z = 24$