Solved Exercises on Solving Two-Step Equations in Grade 7

Master solving two-step equations: addition/subtraction and multiplication/division, with real-world applications through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Addition and Multiplication
Exercise 1
Solve: 2x + 5 = 17
Definition:

Two-step equation: An equation that requires two operations to isolate the variable.

Two-step equation method:
  1. Undo addition or subtraction first (using inverse operation)
  2. Undo multiplication or division next (using inverse operation)
  3. Isolate the variable
  4. Check the solution
Original Equation
2x + 5 = 17
Step 1: Subtract 5 from both sides
2x = 12
Step 2: Divide both sides by 2
x = 6
Step 1: Undo addition

Subtract 5 from both sides to eliminate the constant term:

2x + 5 - 5 = 17 - 5

2x = 12

Step 2: Undo multiplication

Divide both sides by 2 to isolate x:

2x ÷ 2 = 12 ÷ 2

x = 6

Step 3: Check the solution

Substitute x = 6 back into the original equation:

2(6) + 5 = 12 + 5 = 17 ✓

x = 6
Final answer:

x = 6

Applied rules:

• Order of operations reversal: Undo in reverse order of operations

• Inverse operations: Use opposite operations to isolate the variable

• Balancing: Perform the same operation on both sides

2 Subtraction and Multiplication
Exercise 2
Solve: 3x - 8 = 10
Definition:

Subtraction and multiplication: Undo subtraction first, then undo multiplication.

Original Equation
3x - 8 = 10
Step 1: Add 8 to both sides
3x = 18
Step 2: Divide both sides by 3
x = 6
Step 1: Undo subtraction

Add 8 to both sides to eliminate the constant term:

3x - 8 + 8 = 10 + 8

3x = 18

Step 2: Undo multiplication

Divide both sides by 3 to isolate x:

3x ÷ 3 = 18 ÷ 3

x = 6

Step 3: Check the solution

Substitute x = 6 back into the original equation:

3(6) - 8 = 18 - 8 = 10 ✓

x = 6
Final answer:

x = 6

Applied rules:

• Addition inverse: Add to undo subtraction

• Division inverse: Divide to undo multiplication

• Order importance: Undo addition/subtraction before multiplication/division

3 Addition and Division
Exercise 3
Solve: x/4 + 3 = 7
Definition:

Addition and division: Undo addition first, then undo division by multiplying.

Original Equation
x/4 + 3 = 7
Step 1: Subtract 3 from both sides
x/4 = 4
Step 2: Multiply both sides by 4
x = 16
Step 1: Undo addition

Subtract 3 from both sides to eliminate the constant term:

x/4 + 3 - 3 = 7 - 3

x/4 = 4

Step 2: Undo division

Multiply both sides by 4 to isolate x:

(x/4) × 4 = 4 × 4

x = 16

Step 3: Check the solution

Substitute x = 16 back into the original equation:

16/4 + 3 = 4 + 3 = 7 ✓

x = 16
Final answer:

x = 16

Applied rules:

• Subtraction inverse: Subtract to undo addition

• Multiplication inverse: Multiply to undo division

• Sequential undoing: Work in reverse order of operations

Two-Step Equations Rules and Methods
ax + b = c \Rightarrow x = \frac{c - b}{a}
General Two-Step Equation
Addition/Multiplication
ax + b = c → x = (c - b)/a
Undo: subtract b, then divide by a
Subtraction/Multiplication
ax - b = c → x = (c + b)/a
Undo: add b, then divide by a
Addition/Division
x/a + b = c → x = a(c - b)
Undo: subtract b, then multiply by a
Key definitions:

Two-step equation: An algebraic equation requiring exactly two operations to solve for the variable.

Inverse operations: Operations that undo each other (addition/subtraction, multiplication/division).

Order of operations reversal: Undo operations in the reverse order of the order of operations (PEMDAS).

Variable isolation: The process of getting the variable alone on one side of the equation.

Solution verification: Checking the answer by substituting it back into the original equation.

Solving methodology:
  1. Identify operations: Determine what operations are being performed on the variable
  2. Plan sequence: Decide the order to undo operations (reverse order of operations)
  3. Undo addition/subtraction: First eliminate constants using inverse operations
  4. Undo multiplication/division: Then eliminate coefficients using inverse operations
  5. Verify solution: Substitute back to check
Tip 1: Always undo addition/subtraction before multiplication/division.
Tip 2: Think of it as "unpacking" - reverse the order of what was done to the variable.
Tip 3: Perform the same operation on both sides to maintain equality.
Tip 4: Always check your solution by substituting it back into the original equation.
Common errors: Undoing operations in the wrong order, forgetting to apply operations to both sides, making arithmetic mistakes, not verifying the solution.
Exam preparation: Practice all combinations of operations, master inverse operations, develop systematic solving approaches, work with negative numbers and fractions.
Essential rules:

• Addition/Multiplication: ax + b = c → subtract b, then divide by a

• Subtraction/Multiplication: ax - b = c → add b, then divide by a

• Addition/Division: x/a + b = c → subtract b, then multiply by a

• Subtraction/Division: x/a - b = c → add b, then multiply by a

• Balance principle: Whatever you do to one side, do to the other

Solution: Exercises 4 to 5
4 Negative Coefficients
Exercise 4
Solve: -2x + 7 = 15
Definition:

Equations with negative coefficients: The same solving principles apply, but be careful with signs.

Original Equation
-2x + 7 = 15
Step 1: Subtract 7 from both sides
-2x = 8
Step 2: Divide both sides by -2
x = -4
Step 1: Undo addition

Subtract 7 from both sides:

-2x + 7 - 7 = 15 - 7

-2x = 8

Step 2: Undo multiplication

Divide both sides by -2:

-2x ÷ (-2) = 8 ÷ (-2)

x = -4

Step 3: Check the solution

Substitute x = -4 back into the original equation:

-2(-4) + 7 = 8 + 7 = 15 ✓

x = -4
Final answer:

x = -4

Applied rules:

• Negative coefficient: Divide by negative number to isolate variable

• Sign handling: Dividing by negative changes sign of result

• Verification: Always check with negative solutions

5 Real-World Application
Exercise 5
A gym membership costs $20 per month plus a $40 sign-up fee. If someone paid $120 total, how many months did they pay for?
Definition:

Real-world applications: Translating word problems into two-step equations to solve practical problems.

Problem Setup
Let x = number of months
Write Equation
20x + 40 = 120
Solve
x = 4
Step 1: Define the variable

Let x = the number of months (unknown)

Step 2: Set up the equation

Monthly cost × months + sign-up fee = total cost

20x + 40 = 120

Step 3: Solve the equation

Subtract 40 from both sides: 20x = 80

Divide both sides by 20: x = 4

Step 4: Interpret the solution

The person paid for 4 months

Step 5: Verify the solution

Check: 20(4) + 40 = 80 + 40 = 120 ✓

x = 4 (4 months)
Final answer:

The person paid for 4 months

Applied rules:

• Problem translation: Convert real-world situations into mathematical equations

• Variable definition: Clearly define what the variable represents

• Solution interpretation: Relate the mathematical solution back to the real-world context

Detailed Two-Step Equations Guide
ax + b = c \Rightarrow x = \frac{c - b}{a}
General Two-Step Equation Solution
Key definitions:

Two-step equation: An algebraic equation that requires exactly two arithmetic operations to isolate the variable. These equations typically have the form ax + b = c or x/a + b = c, where a, b, and c are constants and x is the variable.

Order of operations reversal: When solving equations, undo operations in the reverse order of how they were applied. Since order of operations (PEMDAS) prioritizes multiplication/division before addition/subtraction, we undo addition/subtraction first.

Inverse operations: Operations that "undo" each other. Addition and subtraction are inverse operations; multiplication and division are inverse operations.

Balance principle: An equation remains true when the same operation is performed on both sides. This principle maintains the equality of the equation throughout the solving process.

Complete solving methodology:
  1. Identify the operations: Determine what operations are being performed on the variable (and in what order)
  2. Plan the sequence: Decide the order to undo operations (reverse order of operations)
  3. Undo addition/subtraction: First eliminate constant terms using inverse operations
  4. Undo multiplication/division: Then eliminate coefficients using inverse operations
  5. Isolate the variable: Get the variable alone on one side of the equation
  6. Verify the solution: Substitute the answer back into the original equation
Tip 1: Think of solving as "unwrapping" the variable - reverse the order of what was done to it.
Tip 2: Always undo addition/subtraction BEFORE multiplication/division.
Tip 3: Keep the equation balanced by doing the same thing to both sides.
Tip 4: Check your answer by substituting it back into the original equation.
Common errors: Undoing operations in the wrong order, forgetting to apply operations to both sides, making sign errors with negative numbers, not checking solutions, misidentifying the operations being performed on the variable.
Applications: Calculating costs with fixed fees, determining time given rate and total amount, solving distance-rate-time problems, working with unit prices, budget planning, and preparing foundation for multi-step equations and more complex algebra.
Essential solving rules:

• Addition/Multiplication: ax + b = c → subtract b, then divide by a

• Subtraction/Multiplication: ax - b = c → add b, then divide by a

• Addition/Division: x/a + b = c → subtract b, then multiply by a

• Subtraction/Division: x/a - b = c → add b, then multiply by a

• Balance principle: Always perform the same operation on both sides

Two-Step Equations Guide

📊
Addition/Multiplication

2x + 5 = 13

Step 1: Subtract 5 → 2x = 8

Step 2: Divide by 2 → x = 4

Subtraction/Multiplication

3x - 7 = 14

Step 1: Add 7 → 3x = 21

Step 2: Divide by 3 → x = 7

Addition/Division

x/4 + 2 = 6

Step 1: Subtract 2 → x/4 = 4

Step 2: Multiply by 4 → x = 16

Order of Operations

Undo in reverse order:

First: Addition/Subtraction

Then: Multiplication/Division

Questions & Answers

Question: Why do I have to undo addition/subtraction before multiplication/division? Can't I do it in any order?

Answer: You must undo operations in reverse order because of how the equation was built! Think of it like unwrapping a gift:

Building the equation: In 2x + 5 = 17

  • Start with x
  • Multiply by 2: 2x
  • Add 5: 2x + 5
  • Equals 17: 2x + 5 = 17

Unwrapping (solving):

  • Undo the last operation first: subtract 5
  • Then undo the earlier operation: divide by 2

Why this matters: If you try to divide first in 2x + 5 = 17:

  • (2x + 5) ÷ 2 = 17 ÷ 2
  • x + 2.5 = 8.5 (This doesn't simplify the problem!)

Undo in reverse order: 17 - 5 = 12, then 12 ÷ 2 = 6

Question: What if I have a negative number in the equation? Like -3x + 8 = 2?

Answer: The same rules apply! The negative sign is part of the coefficient, so you still follow the same order.

For -3x + 8 = 2:

  1. Step 1: Undo addition first: subtract 8 from both sides
  2. -3x + 8 - 8 = 2 - 8
  3. -3x = -6
  1. Step 2: Undo multiplication: divide both sides by -3
  2. -3x ÷ (-3) = -6 ÷ (-3)
  3. x = 2

Check: -3(2) + 8 = -6 + 8 = 2 ✓

Key points:

  • Treat the negative sign as part of the coefficient
  • Dividing by a negative number changes the sign of the result
  • Still follow the same order: undo addition/subtraction first

Question: How do I know if I solved the equation correctly? Is there an easy way to check?

Answer: Yes! Always substitute your answer back into the ORIGINAL equation and see if both sides are equal.

Example: If you solved 2x + 5 = 17 and got x = 6

  • Substitute x = 6 into the original equation:
  • Left side: 2(6) + 5 = 12 + 5 = 17
  • Right side: 17
  • Since 17 = 17, your answer is correct!

Why this works:

  • Your solution should make the original equation true
  • If both sides are equal after substitution, your answer is correct
  • This catches arithmetic errors and method errors

Quick check example:

  • Equation: 3x - 4 = 14, Answer: x = 6
  • Check: 3(6) - 4 = 18 - 4 = 14 ✓

Checking takes only seconds but prevents wrong answers!