Solved Exercises on Solving One-Step Equations in Grade 7

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

Solution: Exercises 1 to 3
1 Addition Equation
Exercise 1
Solve: x + 7 = 12
Definition:

One-step equation: An equation that requires only one operation to isolate the variable.

Addition equation method:
  1. Identify the operation being performed on the variable
  2. Perform the inverse operation on both sides
  3. Isolate the variable
  4. Check the solution
Original Equation
x + 7 = 12
Subtract 7 from both sides
x = 12 - 7
Solution
x = 5
Step 1: Identify the operation

In x + 7 = 12, 7 is being added to x

Step 2: Apply inverse operation

The inverse of addition is subtraction, so subtract 7 from both sides

x + 7 - 7 = 12 - 7

Step 3: Simplify

x = 5

Step 4: Check the solution

Substitute x = 5 back into the original equation:

5 + 7 = 12 ✓

x = 5
Final answer:

x = 5

Applied rules:

• Inverse operations: Addition and subtraction are inverse operations

• Balancing: Whatever you do to one side, do to the other

• Isolation: Get the variable alone on one side

2 Subtraction Equation
Exercise 2
Solve: x - 9 = 4
Definition:

Subtraction equation: An equation where a number is subtracted from the variable. Use addition to solve.

Original Equation
x - 9 = 4
Add 9 to both sides
x = 4 + 9
Solution
x = 13
Step 1: Identify the operation

In x - 9 = 4, 9 is being subtracted from x

Step 2: Apply inverse operation

The inverse of subtraction is addition, so add 9 to both sides

x - 9 + 9 = 4 + 9

Step 3: Simplify

x = 13

Step 4: Check the solution

Substitute x = 13 back into the original equation:

13 - 9 = 4 ✓

x = 13
Final answer:

x = 13

Applied rules:

• Inverse operations: Subtraction and addition are inverse operations

• Balancing: Maintain equality by doing the same to both sides

• Verification: Always check your solution by substituting back

3 Multiplication Equation
Exercise 3
Solve: 3x = 21
Definition:

Multiplication equation: An equation where the variable is multiplied by a number. Use division to solve.

Original Equation
3x = 21
Divide both sides by 3
x = 21 ÷ 3
Solution
x = 7
Step 1: Identify the operation

In 3x = 21, x is being multiplied by 3

Step 2: Apply inverse operation

The inverse of multiplication is division, so divide both sides by 3

3x ÷ 3 = 21 ÷ 3

Step 3: Simplify

x = 7

Step 4: Check the solution

Substitute x = 7 back into the original equation:

3(7) = 21 ✓

x = 7
Final answer:

x = 7

Applied rules:

• Inverse operations: Multiplication and division are inverse operations

• Division principle: Dividing both sides by the coefficient isolates the variable

• Equality maintenance: Operations must be applied to both sides

One-Step Equations Rules and Methods
x + a = b \Rightarrow x = b - a
Addition Equation
Addition
x + a = b → x = b - a
Inverse: subtraction
Subtraction
x - a = b → x = b + a
Inverse: addition
Multiplication
ax = b → x = b ÷ a
Inverse: division
Key definitions:

Equation: A mathematical statement showing that two expressions are equal.

Variable: A symbol (usually a letter) that represents an unknown value.

Solution: The value of the variable that makes the equation true.

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

One-step equation: An equation that requires only one operation to solve.

Solving methodology:
  1. Identify the equation type: Determine the operation on the variable
  2. Select the inverse operation: Choose the operation that will isolate the variable
  3. Apply to both sides: Perform the operation on both sides of the equation
  4. Solve for the variable: Simplify to get the variable alone
  5. Verify the solution: Substitute back to check
Tip 1: Always perform the same operation on both sides to keep the equation balanced.
Tip 2: Use inverse operations to "undo" what's being done to the variable.
Tip 3: Always check your answer by substituting it back into the original equation.
Tip 4: Remember: addition and subtraction are inverses; multiplication and division are inverses.
Common errors: Forgetting to apply operations to both sides, using the wrong inverse operation, not checking the solution, misapplying the inverse operation.
Exam preparation: Practice all four operation types, master inverse operations, learn to verify solutions quickly, work with negative numbers and fractions.
Essential rules:

• Addition equation: x + a = b → subtract a from both sides

• Subtraction equation: x - a = b → add a to both sides

• Multiplication equation: ax = b → divide both sides by a

• Division equation: x/a = b → multiply both sides by a

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

Solution: Exercises 4 to 5
4 Division Equation
Exercise 4
Solve: x/4 = 8
Definition:

Division equation: An equation where the variable is divided by a number. Use multiplication to solve.

Original Equation
x/4 = 8
Multiply both sides by 4
x = 8 × 4
Solution
x = 32
Step 1: Identify the operation

In x/4 = 8, x is being divided by 4

Step 2: Apply inverse operation

The inverse of division is multiplication, so multiply both sides by 4

(x/4) × 4 = 8 × 4

Step 3: Simplify

x = 32

Step 4: Check the solution

Substitute x = 32 back into the original equation:

32/4 = 8 ✓

x = 32
Final answer:

x = 32

Applied rules:

• Inverse operations: Division and multiplication are inverse operations

• Multiplication principle: Multiplying both sides by the divisor isolates the variable

• Verification: Always substitute back to verify your solution

5 Real-World Application
Exercise 5
Sarah bought 5 identical books for a total of $45. Write and solve an equation to find the cost of each book.
Definition:

Real-world applications: Translating word problems into mathematical equations and solving them to find unknown values.

Problem Setup
Let x = cost of each book
Write Equation
5x = 45
Solve
x = 9
Step 1: Define the variable

Let x = the cost of each book (unknown)

Step 2: Set up the equation

5 books × cost per book = total cost

5x = 45

Step 3: Solve the equation

Divide both sides by 5: x = 45 ÷ 5

x = 9

Step 4: Interpret the solution

Each book costs $9

Step 5: Verify the solution

5 books × $9 each = $45 ✓

x = 9 (each book costs $9)
Final answer:

Each book costs $9

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 problem

Detailed One-Step Equations Guide
x + a = b \Rightarrow x = b - a
Addition Equation Solution
Key definitions:

One-step equation: An algebraic equation that requires only one arithmetic operation to solve for the variable. These equations have the form: variable ± number = number or variable × number = number or variable ÷ number = number.

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

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

Solution verification: The process of substituting the found value back into the original equation to ensure both sides are equal.

Complete solving methodology:
  1. Identify the equation type: Determine whether addition, subtraction, multiplication, or division is being performed on the variable
  2. Select the inverse operation: Choose the operation that will cancel out what's being done to the variable
  3. Apply operation to both sides: Perform the inverse operation on both sides of the equation to maintain balance
  4. Isolate the variable: Simplify to get the variable alone on one side
  5. Solve for the value: Complete the arithmetic to find the variable's value
  6. Verify the solution: Substitute the answer back into the original equation to confirm it works
Tip 1: Think of the equation like a balanced scale - whatever you do to one side, you must do to the other to keep it balanced.
Tip 2: To isolate the variable, use the inverse operation: if something is added, subtract it; if multiplied, divide by it.
Tip 3: Always check your solution by substituting it back into the original equation - this catches most errors.
Tip 4: When dividing by a fraction, multiply by its reciprocal (e.g., dividing by 1/2 is the same as multiplying by 2).
Common errors: Forgetting to apply operations to both sides, using the wrong inverse operation, not properly distributing negative signs, failing to check solutions, misreading the equation operation.
Applications: Solving real-world problems involving rates, proportions, pricing, distances, time calculations, measurement conversions, and preparing foundation for multi-step equations and more complex algebra.
Essential solving rules:

• Addition equation (x + a = b): Subtract a from both sides

• Subtraction equation (x - a = b): Add a to both sides

• Multiplication equation (ax = b): Divide both sides by a

• Division equation (x/a = b): Multiply both sides by a

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

One-Step Equations Guide

📊
Addition

x + 5 = 12

Subtract 5 from both sides

x = 12 - 5 = 7

Subtraction

x - 3 = 8

Add 3 to both sides

x = 8 + 3 = 11

Multiplication

4x = 20

Divide both sides by 4

x = 20 ÷ 4 = 5

Division

x/6 = 3

Multiply both sides by 6

x = 3 × 6 = 18

Questions & Answers

Question: Why do I need to do the same thing to both sides of the equation? Can't I just do it to one side?

Answer: Think of an equation like a balanced scale. If you have 5 pounds on each side, the scale is balanced. If you remove 2 pounds from one side only, the scale becomes unbalanced!

Example: In x + 3 = 7

  • LEFT side: x + 3
  • RIGHT side: 7
  • These sides are equal: x + 3 = 7

If we only subtract 3 from the left:

  • LEFT: x + 3 - 3 = x
  • RIGHT: still 7
  • Now x = 7, which is wrong!

If we subtract 3 from both sides:

  • LEFT: x + 3 - 3 = x
  • RIGHT: 7 - 3 = 4
  • Now x = 4, which is correct!

The balance principle keeps the equation true!

Question: How do I know whether to add, subtract, multiply, or divide? It's confusing!

Answer: Look at what's being done to the variable and use the OPPOSITE (inverse) operation!

Easy pattern to remember:

  • If something is ADDED to the variable: SUBTRACT it from both sides
  • If something is SUBTRACTED from the variable: ADD it to both sides
  • If the variable is MULTIPLIED by a number: DIVIDE both sides by that number
  • If the variable is DIVIDED by a number: MULTIPLY both sides by that number

Examples:

  • x + 5 = 12 → subtract 5 (opposite of adding)
  • x - 3 = 8 → add 3 (opposite of subtracting)
  • 6x = 18 → divide by 6 (opposite of multiplying)
  • x/4 = 5 → multiply by 4 (opposite of dividing)

The inverse operation "undoes" what's been done to the variable!

Question: Why do I need to check my answer? Isn't solving the equation enough?

Answer: Checking your answer is like proofreading your work - it catches mistakes and confirms your solution is correct!

Why checking is important:

  • Catches arithmetic errors: Adding wrong, forgetting negative signs, etc.
  • Verifies logic: Confirms your method was correct
  • Builds confidence: You know your answer is right
  • Develops good habits: Essential for more complex equations

How to check:

  • Solve: x + 8 = 15 → x = 7
  • Check: Substitute x = 7 into original equation
  • 7 + 8 = 15 ✓ (True! Your answer is correct)

Example of catching an error:

  • You solve: x - 4 = 10 and get x = 6
  • Check: 6 - 4 = 2, but original says x - 4 = 10
  • Since 2 ≠ 10, you know x = 6 is wrong and need to fix it

Checking takes only seconds but saves you from wrong answers!