Solved Exercises on One-Step Multiplication & Division Equations in Pre-algebra

Master one-step multiplication and division equations: understanding how to solve simple equations with multiplication and division operations through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Multiplication equation
Exercise 1
Solve: \(3x = 15\)
Definition:

Multiplication equation: An equation where the variable is multiplied by a number. To solve, divide both sides by that number to isolate the variable.

Solving method:
  1. Identify the operation (multiplication in this case)
  2. Apply the inverse operation (division) to both sides
  3. Simplify both sides
  4. Verify the solution
Original
\(3x = 15\)
Divide by 3
\(x = 5\)
Check
\(3 \times 5 = 15\) ✓
Step 1: Identify the operation

The variable \(x\) is multiplied by 3, so we need to divide both sides by 3

Step 2: Apply inverse operation

Divide both sides by 3: \(\frac{3x}{3} = \frac{15}{3}\)

Step 3: Simplify

Left side: \(\frac{3x}{3} = x\)

Right side: \(\frac{15}{3} = 5\)

So: \(x = 5\)

Step 4: Verify the solution

Substitute \(x = 5\) back into the original equation: \(3 \times 5 = 15\) ✓

\(x = 5\)
Final answer:

\(x = 5\)

Applied rules:

• Inverse operations: Division undoes multiplication

• Balance principle: Perform the same operation on both sides

• Verification: Always check your solution

2 Division equation
Exercise 2
Solve: \(\frac{y}{4} = 7\)
Definition:

Division equation: An equation where the variable is divided by a number. To solve, multiply both sides by that number to isolate the variable.

Original
\(\frac{y}{4} = 7\)
Multiply by 4
\(y = 28\)
Check
\(\frac{28}{4} = 7\) ✓
Step 1: Identify the operation

The variable \(y\) is divided by 4, so we need to multiply both sides by 4

Step 2: Apply inverse operation

Multiply both sides by 4: \(\frac{y}{4} \times 4 = 7 \times 4\)

Step 3: Simplify

Left side: \(\frac{y}{4} \times 4 = y\)

Right side: \(7 \times 4 = 28\)

So: \(y = 28\)

Step 4: Verify the solution

Substitute \(y = 28\) back into the original equation: \(\frac{28}{4} = 7\) ✓

\(y = 28\)
Final answer:

\(y = 28\)

Applied rules:

• Inverse operations: Multiplication undoes division

• Balance principle: Perform the same operation on both sides

• Verification: Always check your solution

3 Multiplication with negative number
Exercise 3
Solve: \(-2z = 14\)
Definition:

Multiplication with negative number: An equation where the variable is multiplied by a negative number. To solve, divide both sides by that negative number.

Original
\(-2z = 14\)
Divide by -2
\(z = -7\)
Check
\(-2 \times (-7) = 14\) ✓
Step 1: Identify the operation

The variable \(z\) is multiplied by \(-2\), so we need to divide both sides by \(-2\)

Step 2: Apply inverse operation

Divide both sides by \(-2\): \(\frac{-2z}{-2} = \frac{14}{-2}\)

Step 3: Simplify

Left side: \(\frac{-2z}{-2} = z\)

Right side: \(\frac{14}{-2} = -7\)

So: \(z = -7\)

Step 4: Verify the solution

Substitute \(z = -7\) back into the original equation: \(-2 \times (-7) = 14\) ✓

\(z = -7\)
Final answer:

\(z = -7\)

Applied rules:

• Division with negatives: Dividing by a negative flips the sign

• Inverse operations: Division undoes multiplication

• Balance principle: Perform the same operation on both sides

Rules and methods, laws,...
\(ax = b\)
Multiplication Equation
Multiplication
\(ax = b\)
Solution: \(x = \frac{b}{a}\)
Division
\(\frac{x}{a} = b\)
Solution: \(x = ab\)
Negative Multiplication
\(-ax = b\)
Solution: \(x = -\frac{b}{a}\)
Negative Division
\(\frac{x}{-a} = b\)
Solution: \(x = -ab\)
Key definitions:

One-Step Multiplication Equation: An equation of the form \(ax = b\) where \(a\) and \(b\) are constants. The solution is found by dividing both sides by \(a\): \(x = \frac{b}{a}\).

One-Step Division Equation: An equation of the form \(\frac{x}{a} = b\) where \(a\) and \(b\) are constants. The solution is found by multiplying both sides by \(a\): \(x = ab\).

Inverse Operations: Operations that undo each other. Multiplication and division are inverse operations.

Balance Principle: The fundamental concept that both sides of an equation must remain equal. Whatever is done to one side must be done to the other side.

Verification: The process of checking that a solution is correct by substituting it back into the original equation.

Complete methodology:
  1. Identify the equation type: Determine if it's multiplication or division
  2. Identify the operation: See what operation is being performed on the variable
  3. Select the inverse operation: Use division to undo multiplication, multiplication to undo division
  4. Apply to both sides: Perform the inverse operation on both sides of the equation
  5. Simplify: Calculate the result
  6. Verify: Check your solution by substituting back into the original equation
Tip 1: Remember: Multiplication and division are inverse operations
Tip 2: Always perform the same operation on both sides
Tip 3: When dividing by a negative, the sign of the result changes
Tip 4: Always verify your solution by substituting back
Common errors: Forgetting to apply operations to both sides, using the wrong inverse operation, making sign errors with negative numbers, not checking the solution.
Memory aids: "What you do to one side, you do to the other", "Multiplication and division are opposites", "Undo the operation to isolate the variable".
Solution: Exercises 4 to 5
4 Division with negative number
Exercise 4
Solve: \(\frac{w}{-3} = 9\)
Definition:

Division with negative number: An equation where the variable is divided by a negative number. To solve, multiply both sides by that negative number.

Original
\(\frac{w}{-3} = 9\)
Multiply by -3
\(w = -27\)
Check
\(\frac{-27}{-3} = 9\) ✓
Step 1: Identify the operation

The variable \(w\) is divided by \(-3\), so we need to multiply both sides by \(-3\)

Step 2: Apply inverse operation

Multiply both sides by \(-3\): \(\frac{w}{-3} \times (-3) = 9 \times (-3)\)

Step 3: Simplify

Left side: \(\frac{w}{-3} \times (-3) = w\)

Right side: \(9 \times (-3) = -27\)

So: \(w = -27\)

Step 4: Verify the solution

Substitute \(w = -27\) back into the original equation: \(\frac{-27}{-3} = 9\) ✓

\(w = -27\)
Final answer:

\(w = -27\)

Applied rules:

• Multiplication with negatives: Multiplying by a negative flips the sign

• Inverse operations: Multiplication undoes division

• Balance principle: Perform the same operation on both sides

5 Real-world application
Exercise 5
A rectangular garden has an area of 24 square feet. If the width is 4 feet, what is the length? Write and solve the equation.
Definition:

Real-world application: Translating word problems into mathematical equations helps connect abstract concepts to practical situations.

Set up
\(4l = 24\)
Divide by 4
\(l = 6\)
Check
\(4 \times 6 = 24\) ✓
Step 1: Define the variable

Let \(l\) = the length of the garden in feet

Step 2: Set up the equation

Area = Length × Width, so: \(l \times 4 = 24\), or \(4l = 24\)

Step 3: Solve the equation

Divide both sides by 4: \(\frac{4l}{4} = \frac{24}{4}\), so \(l = 6\)

Step 4: Verify the solution

If the length is 6 feet and the width is 4 feet, the area is \(6 \times 4 = 24\) square feet ✓

\(l = 6\)
Final answer:

The length of the garden is 6 feet. The equation is \(4l = 24\), and the solution is \(l = 6\).

Applied rules:

• Word problem translation: Convert words into mathematical expressions

• Multiplication equation: Divide both sides to solve

• Verification: Check that solution makes sense in context

Comprehensive Summary: One-Step Multiplication & Division Equations
\(ax = b \quad \text{or} \quad \frac{x}{a} = b\)
General Forms
Key definitions:

One-Step Multiplication Equation: An equation of the form \(ax = b\) where the variable is multiplied by a constant. The solution is found by dividing both sides by the coefficient: \(x = \frac{b}{a}\).

One-Step Division Equation: An equation of the form \(\frac{x}{a} = b\) where the variable is divided by a constant. The solution is found by multiplying both sides by the divisor: \(x = ab\).

Inverse Operations: Operations that undo each other. Multiplication and division are inverse operations: multiplying by a number and then dividing by the same number brings you back to the original value.

Balance Principle: The fundamental concept that both sides of an equation must remain equal. Whatever is done to one side must be done to the other side to maintain equality.

Isolation: The process of getting the variable by itself on one side of the equation through inverse operations.

Complete methodology:
  1. Identify the equation type: Determine if it's multiplication (\(ax = b\)) or division (\(\frac{x}{a} = b\))
  2. Identify the operation on the variable: See what operation is being performed with the variable
  3. Select the inverse operation: Use division to undo multiplication, multiplication to undo division
  4. Apply to both sides: Perform the inverse operation on both sides of the equation
  5. Simplify: Calculate the result to isolate the variable
  6. Verify: Check your solution by substituting back into the original equation
Tip 1: Remember that dividing by a negative number changes the sign of the result
Tip 2: Always keep the variable on one side and move numbers to the other side
Tip 3: Picture the equation as a balanced scale - both sides must stay equal
Tip 4: Always verify your answer by substituting it back into the original equation
Tip 5: When solving, focus on isolating the variable completely
Common errors: Forgetting to apply operations to both sides, using the wrong inverse operation, making sign errors with negative numbers, not verifying the solution.
Memory aids: "Multiplication and division are opposites", "What you do to one side, you do to the other", "Undo the operation to get the variable alone".
Essential rules to remember:

• Balance rule: Whatever you do to one side of the equation, you must do to the other side

• Inverse operations: Multiplication and division are inverses of each other

• Solution for multiplication: For \(ax = b\), the solution is \(x = \frac{b}{a}\)

• Solution for division: For \(\frac{x}{a} = b\), the solution is \(x = ab\)

• Verification: Always check your solution by substituting back into the original equation

Visualization: Multiplication & Division Relationships
Exercise 6: Equation Visualization
Visualizing how multiplication and division equations work:
\(f_1(x) = 3x\) and \(g_1(x) = 15\) intersect at \(x = 5\)
\(f_2(x) = \frac{x}{4}\) and \(g_2(x) = 7\) intersect at \(x = 28\)
\(f_3(x) = -2x\) and \(g_3(x) = 14\) intersect at \(x = -7\)

Analysis: The chart shows how equations represent balanced relationships.

  • Multiplication equations create linear functions
  • Division equations also create linear functions
  • Solutions occur where functions intersect
  • Each type maintains the balance principle

Questions & Answers

Question: I get confused about when to multiply and when to divide. How do I remember which one to use?

Answer: You use the inverse operation to undo what was done to the variable:

Rule: 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:

  • \(4x = 20\): Variable is multiplied by 4, so divide both sides by 4: \(x = 5\)
  • \(\frac{x}{3} = 8\): Variable is divided by 3, so multiply both sides by 3: \(x = 24\)
  • \(-5x = 15\): Variable is multiplied by -5, so divide both sides by -5: \(x = -3\)
  • \(\frac{x}{-2} = 6\): Variable is divided by -2, so multiply both sides by -2: \(x = -12\)

Remember: "Undo the operation" - if multiplication was done, use division to undo it. If division was done, use multiplication to undo it.

Question: Why do I need to do the same thing to both sides of the equation? Can't I just move numbers around?

Answer: An equation represents a balance. Think of it like a physical scale:

If you have \(3x = 15\), both sides are equal (balanced).

If you divide only the left side by 3, you get \(x = 15\), but now the scale is unbalanced because \(x\) doesn't equal 15!

To keep the balance:

  • Left side: \(\frac{3x}{3} = x\)
  • Right side: \(\frac{15}{3} = 5\)
  • Now: \(x = 5\) (balanced again!)

The equals sign (=) means "these two sides are equal." To keep them equal, whatever you do to one side, you must do to the other side.

This is the fundamental principle of all algebraic equations and ensures that your solution is valid.

Question: What happens when I divide by a negative number? I always get confused about the signs.

Answer: When you divide by a negative number, the sign of the result changes:

Sign rules:

  • Positive ÷ Negative = Negative
  • Negative ÷ Negative = Positive
  • Positive ÷ Positive = Positive
  • Negative ÷ Positive = Negative

Examples:

  • \(-2x = 14\): Divide both sides by -2: \(x = \frac{14}{-2} = -7\)
  • \(-3x = -12\): Divide both sides by -3: \(x = \frac{-12}{-3} = 4\)
  • \(\frac{x}{-4} = 5\): Multiply both sides by -4: \(x = 5 \times (-4) = -20\)

Remember: Dividing by a negative number flips the sign of the result. Always double-check your work by substituting back into the original equation.