Solved Exercises on Two-Step Equations in Grade 8

Master two-step equations: addition/subtraction with multiplication/division, with decimals, with fractions, and word problems through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Addition and Multiplication
Exercise 1
Solve for \(x\):
\(3x + 7 = 22\)
Definition:

Two-Step Equation: An equation requiring two operations to isolate the variable

Two-Step Method:
  1. Undo addition or subtraction first (inverse operation)
  2. Undo multiplication or division second (inverse operation)
  3. Simplify to find the variable
  4. Verify the solution by substituting back
Original Equation
\(3x + 7 = 22\)
Subtract 7 from both sides
\(3x + 7 - 7 = 22 - 7\)
Simplify
\(3x = 15\)
Divide both sides by 3
\(\frac{3x}{3} = \frac{15}{3}\)
Final Solution
\(x = 5\)
Step 1: Undo addition

The variable \(x\) is first multiplied by 3, then 7 is added. Undo addition first: \(3x + 7 - 7 = 22 - 7\)

Step 2: Simplify

Left side: \(3x + 7 - 7 = 3x\)

Right side: \(22 - 7 = 15\)

So: \(3x = 15\)

Step 3: Undo multiplication

To undo multiplying by 3, divide both sides by 3: \(\frac{3x}{3} = \frac{15}{3}\)

Step 4: Simplify

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

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

Step 5: Write the solution

\(x = 5\)

Step 6: Verify the solution

Substitute \(x = 5\) back into the original equation: \(3(5) + 7 = 15 + 7 = 22\) ✓

\(x = 5\)
Final answer:

\(x = 5\)

Applied rules:

• Order of Operations (Reverse): Undo operations in reverse order

• Inverse Operations: Subtraction undoes addition, division undoes multiplication

• Balanced Equation: Whatever you do to one side, do to the other

2 Subtraction and Division
Exercise 2
Solve for \(y\):
\(\frac{y}{4} - 6 = 3\)
Definition:

Two-Step Division Equation: An equation involving division and addition/subtraction

Original Equation
\(\frac{y}{4} - 6 = 3\)
Add 6 to both sides
\(\frac{y}{4} - 6 + 6 = 3 + 6\)
Simplify
\(\frac{y}{4} = 9\)
Multiply both sides by 4
\(\frac{y}{4} \times 4 = 9 \times 4\)
Final Solution
\(y = 36\)
Step 1: Undo subtraction

The variable \(y\) is first divided by 4, then 6 is subtracted. Undo subtraction first: \(\frac{y}{4} - 6 + 6 = 3 + 6\)

Step 2: Simplify

Left side: \(\frac{y}{4} - 6 + 6 = \frac{y}{4}\)

Right side: \(3 + 6 = 9\)

So: \(\frac{y}{4} = 9\)

Step 3: Undo division

To undo dividing by 4, multiply both sides by 4: \(\frac{y}{4} \times 4 = 9 \times 4\)

Step 4: Simplify

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

Right side: \(9 \times 4 = 36\)

Step 5: Write the solution

\(y = 36\)

Step 6: Verify the solution

Substitute \(y = 36\) back into the original equation: \(\frac{36}{4} - 6 = 9 - 6 = 3\) ✓

\(y = 36\)
Final answer:

\(y = 36\)

Applied rules:

• Order of Operations (Reverse): Undo operations in reverse order

• Inverse Operations: Addition undoes subtraction, multiplication undoes division

• Balanced Equation: Whatever you do to one side, do to the other

3 Multiplication and Subtraction
Exercise 3
Solve for \(z\):
\(5z - 12 = 18\)
Definition:

Two-Step Multiplication Equation: An equation involving multiplication and addition/subtraction

Original Equation
\(5z - 12 = 18\)
Add 12 to both sides
\(5z - 12 + 12 = 18 + 12\)
Simplify
\(5z = 30\)
Divide both sides by 5
\(\frac{5z}{5} = \frac{30}{5}\)
Final Solution
\(z = 6\)
Step 1: Undo subtraction

The variable \(z\) is first multiplied by 5, then 12 is subtracted. Undo subtraction first: \(5z - 12 + 12 = 18 + 12\)

Step 2: Simplify

Left side: \(5z - 12 + 12 = 5z\)

Right side: \(18 + 12 = 30\)

So: \(5z = 30\)

Step 3: Undo multiplication

To undo multiplying by 5, divide both sides by 5: \(\frac{5z}{5} = \frac{30}{5}\)

Step 4: Simplify

Left side: \(\frac{5z}{5} = z\)

Right side: \(\frac{30}{5} = 6\)

Step 5: Write the solution

\(z = 6\)

Step 6: Verify the solution

Substitute \(z = 6\) back into the original equation: \(5(6) - 12 = 30 - 12 = 18\) ✓

\(z = 6\)
Final answer:

\(z = 6\)

Applied rules:

• Order of Operations (Reverse): Undo operations in reverse order

• Inverse Operations: Addition undoes subtraction, division undoes multiplication

• Balanced Equation: Whatever you do to one side, do to the other

Two-Step Equations Rules and Methods
\(ax + b = c \Rightarrow x = \frac{c - b}{a}\)
General Form
Form 1
\(ax + b = c\)
Subtract b, then divide by a
Form 2
\(ax - b = c\)
Add b, then divide by a
Form 3
\(\frac{x}{a} + b = c\)
Subtract b, then multiply by a
Form 4
\(\frac{x}{a} - b = c\)
Add b, then multiply by a
Inverse Operations
Addition ↔ Subtraction
Multiplication ↔ Division
Order of Undoing
Last operation first
Reverse order of operations
Key Concepts: Two-step equations require undoing operations in reverse order. Always perform the same operation on both sides to maintain balance.
Verification: Always substitute your solution back into the original equation to check accuracy.
Tip 1: Undo operations in reverse order of the order of operations (PEMDAS backwards).
Tip 2: Always check your answer by substituting it back into the original equation.
Tip 3: Remember that division by zero is undefined, so check that your divisor is not zero.
Solution: Exercises 4 to 5
4 Division and Addition with Decimals
Exercise 4
Solve for \(w\):
\(\frac{w}{2.5} + 4.5 = 9.5\)
Definition:

Two-Step Decimal Equation: An equation involving decimals and requiring two operations

Original Equation
\(\frac{w}{2.5} + 4.5 = 9.5\)
Subtract 4.5 from both sides
\(\frac{w}{2.5} = 9.5 - 4.5\)
Simplify
\(\frac{w}{2.5} = 5\)
Multiply both sides by 2.5
\(w = 5 \times 2.5\)
Final Solution
\(w = 12.5\)
Step 1: Undo addition

The variable \(w\) is first divided by 2.5, then 4.5 is added. Undo addition first: \(\frac{w}{2.5} + 4.5 - 4.5 = 9.5 - 4.5\)

Step 2: Simplify

Right side: \(9.5 - 4.5 = 5\)

So: \(\frac{w}{2.5} = 5\)

Step 3: Undo division

To undo dividing by 2.5, multiply both sides by 2.5: \(\frac{w}{2.5} \times 2.5 = 5 \times 2.5\)

Step 4: Simplify

Left side: \(\frac{w}{2.5} \times 2.5 = w\)

Right side: \(5 \times 2.5 = 12.5\)

Step 5: Write the solution

\(w = 12.5\)

Step 6: Verify the solution

Substitute \(w = 12.5\) back into the original equation: \(\frac{12.5}{2.5} + 4.5 = 5 + 4.5 = 9.5\) ✓

\(w = 12.5\)
Final answer:

\(w = 12.5\)

Applied rules:

• Order of Operations (Reverse): Undo operations in reverse order

• Inverse Operations: Subtraction undoes addition, multiplication undoes division

• Balanced Equation: Whatever you do to one side, do to the other

5 Word Problem with Fractions
Exercise 5
A number is divided by 4 and then increased by 3 to get 8. Find the number.
Definition:

Word Problem: Translate words into mathematical expressions and solve

Let the number be \(n\)
\(\frac{n}{4} + 3 = 8\)
Subtract 3 from both sides
\(\frac{n}{4} = 8 - 3\)
Simplify
\(\frac{n}{4} = 5\)
Multiply both sides by 4
\(n = 5 \times 4\)
Final Solution
\(n = 20\)
Step 1: Define the variable

Let the unknown number be \(n\)

Step 2: Translate the words into an equation

"A number is divided by 4 and then increased by 3 to get 8" becomes: \(\frac{n}{4} + 3 = 8\)

Step 3: Undo addition first

Subtract 3 from both sides: \(\frac{n}{4} + 3 - 3 = 8 - 3\)

Step 4: Simplify

Right side: \(8 - 3 = 5\)

So: \(\frac{n}{4} = 5\)

Step 5: Undo division

Multiply both sides by 4: \(\frac{n}{4} \times 4 = 5 \times 4\)

Step 6: Simplify

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

Right side: \(5 \times 4 = 20\)

Step 7: Write the solution

\(n = 20\)

Step 8: Verify the solution

Check: \(\frac{20}{4} + 3 = 5 + 3 = 8\) ✓

The number is 20
Final answer:

The number is 20

Applied rules:

• Word Translation: Convert verbal statements to algebraic expressions

• Order of Operations (Reverse): Undo operations in reverse order

• Balanced Equation: Whatever you do to one side, do to the other

Complete Guide: Two-Step Equations, Rules, Methods, and Applications
\(ax + b = c \Rightarrow x = \frac{c - b}{a}\)
General Form
Key definitions:

Two-Step Equation: An equation requiring exactly two operations to isolate the variable

Variable: A symbol representing an unknown value

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

Order of Undoing: Reverse the order of operations applied to the variable

Complete methodology:
  1. Identify operations: Determine what operations are being performed on the variable
  2. Order of undoing: Undo operations in reverse order of how they were applied
  3. Apply inverses: Use inverse operations to both sides of the equation
  4. Simplify: Reduce both sides to isolate the variable
  5. Verify: Substitute the solution back into the original equation
Tip 1: Undo operations in reverse order of how they were applied to the variable.
Tip 2: Always perform the same operation on both sides to maintain equality.
Tip 3: Always check your solution by substituting it back into the original equation.
Tip 4: When working with fractions, multiply by the reciprocal to undo division.
Common errors: Performing operations in the wrong order, forgetting to apply operations to both sides, making sign errors with negative numbers.
Real-World Applications: Two-step equations model many real-life situations like calculating costs, distances, and rates.
Essential rules to memorize:

• Order of Undoing: Reverse the order of operations applied to the variable

• Inverse Operations: Addition ↔ Subtraction, Multiplication ↔ Division

• Balanced Equation: Perform same operation on both sides

• Verification: Always check your solution

• General Forms: \(ax + b = c\), \(ax - b = c\), \(\frac{x}{a} + b = c\), \(\frac{x}{a} - b = c\)

Exercise with Visualization: Two-Step Equations Solutions
Exercise 6: Equation Solutions Comparison
Consider the following equations and their solutions:
\(f_1(x) = 3x + 7 = 22 \Rightarrow x = 5\)
\(f_2(y) = \frac{y}{4} - 6 = 3 \Rightarrow y = 36\)
\(f_3(z) = 5z - 12 = 18 \Rightarrow z = 6\)

Analysis: The chart shows how different two-step equations have different solutions.

  • \(3x + 7 = 22\) (solution: \(x = 5\))
  • \(\frac{y}{4} - 6 = 3\) (solution: \(y = 36\))
  • \(5z - 12 = 18\) (solution: \(z = 6\))

Questions & Answers

Question: Why do I have to undo operations in reverse order? Why not just do them in the same order?

Answer: Think of solving an equation like unwrapping a gift that was wrapped in a specific order.

Imagine you wrap a gift by putting it in a box and then wrapping the box with paper:

  • Wrapping: Put gift in box, then wrap box with paper
  • Unwrapping: Remove paper first, then take gift from box (reverse order!)

For the equation \(3x + 7 = 22\), the variable \(x\) went through operations in this order:

  1. First: Multiply by 3
  2. Second: Add 7

To solve, we undo in reverse order:

  1. First: Undo addition (subtract 7)
  2. Second: Undo multiplication (divide by 3)

This is the same principle as the order of operations (PEMDAS) reversed!

Question: How do I know whether to add/subtract first or multiply/divide first?

Answer: Look at what happened to your variable and undo in reverse order:

For \(3x + 7 = 22\):

  1. What happened to \(x\)? First it was multiplied by 3, then 7 was added
  2. Undo in reverse order: First subtract 7, then divide by 3

For \(\frac{y}{4} - 6 = 3\):

  1. What happened to \(y\)? First it was divided by 4, then 6 was subtracted
  2. Undo in reverse order: First add 6, then multiply by 4

Memory aid: Think of your variable as going through a series of operations. List them in order, then reverse the list for your solution steps.

For \(5z - 12 = 18\): \(z\) → multiply by 5 → subtract 12

So undo: add 12 → divide by 5

Question: What if I have fractions in the equation? Like \(\frac{2}{3}x + 4 = 10\)?

Answer: Follow the same steps, but when undoing multiplication by a fraction, multiply by its reciprocal:

For \(\frac{2}{3}x + 4 = 10\):

  1. Undo addition first: \(\frac{2}{3}x + 4 - 4 = 10 - 4\), so \(\frac{2}{3}x = 6\)
  2. Undo multiplication by \(\frac{2}{3}\) by multiplying by its reciprocal \(\frac{3}{2}\): \(\frac{2}{3}x \times \frac{3}{2} = 6 \times \frac{3}{2}\)
  3. Simplify: \(x = \frac{18}{2} = 9\)

Verification: \(\frac{2}{3}(9) + 4 = \frac{18}{3} + 4 = 6 + 4 = 10\) ✓

Remember: To undo multiplication by \(\frac{a}{b}\), multiply by \(\frac{b}{a}\) (the reciprocal).

This works because \(\frac{a}{b} \times \frac{b}{a} = 1\), which leaves just the variable.

Question: How do I solve word problems that lead to two-step equations?

Answer: Follow these steps for word problems:

Step 1: Identify the unknown

What are you trying to find? Assign it a variable.

Step 2: Translate words into math

Look for key phrases:

  • "increased by", "more than" → addition
  • "decreased by", "less than" → subtraction
  • "multiplied by", "times" → multiplication
  • "divided by", "quotient" → division

Step 3: Set up the equation

Express the relationship mathematically.

Step 4: Solve using two-step method

Example: "Three times a number decreased by 5 equals 16"

  • Unknown: let it be \(x\)
  • Translate: "three times a number" = \(3x\), "decreased by 5" = \(-5\), "equals 16" = \(= 16\)
  • Equation: \(3x - 5 = 16\)
  • Solve: Add 5, then divide by 3
  • Solution: \(x = 7\)

Question: How can I check if my answer is correct? What if I made a mistake?

Answer: Always substitute your solution back into the ORIGINAL equation to verify:

Example: Solve \(3x + 7 = 22\), you get \(x = 5\)

  1. Take the original equation: \(3x + 7 = 22\)
  2. Replace \(x\) with your answer (5): \(3(5) + 7 = 22\)
  3. Calculate: \(15 + 7 = 22\), so \(22 = 22\) ✓ (True! Your answer is correct)

If you get something like \(23 = 22\), then your answer is wrong.

Let's say you thought \(x = 6\):

  • Substitute: \(3(6) + 7 = 22\)
  • Calculate: \(18 + 7 = 22\), so \(25 = 22\) ✗ (False! So \(x = 6\) is wrong)

Verification is crucial:

  • Catches calculation errors
  • Confirms your solution is correct
  • Builds confidence in your answer
  • Required by teachers for full credit

Always verify your answers!