Solved Exercises on Two-Step Equations in Integrated Math 1

Master two-step equations: addition/subtraction with multiplication/division, and complex equations through these 5 detailed exercises with comprehensive solutions.

Solution: Exercises 1 to 3
1 Addition and multiplication
Exercise 1
Solve for x:
\(3x + 5 = 20\)
Definition:

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

Order of operations: Work backwards through PEMDAS/BODMAS to undo operations

Inverse operations: Addition/subtraction and multiplication/division are opposites

Two-step solving method:

To solve equations with addition/subtraction AND multiplication/division:

  1. Undo addition/subtraction first: Remove constants from the variable side
  2. Undo multiplication/division second: Isolate the variable
  3. Verify: Substitute solution back into original equation
Original
\(3x + 5 = 20\)
Subtract 5
\(3x = 15\)
Divide by 3
\(x = 5\)
Step 1: Write the equation

\(3x + 5 = 20\)

Step 2: Undo addition by subtracting 5 from both sides

\(3x + 5 - 5 = 20 - 5\)

\(3x = 15\)

Step 3: Undo multiplication by dividing both sides by 3

\(\frac{3x}{3} = \frac{15}{3}\)

\(x = 5\)

Step 4: Verify the solution

Substitute \(x = 5\) into original: \(3(5) + 5 = 15 + 5 = 20\) ✓

\(x = 5\)
Final answer:

The solution is \(x = 5\)

Applied rules:

• Addition/Subtraction Property: Adding or subtracting the same number from both sides maintains equality

• Multiplication/Division Property: Dividing both sides by the same non-zero number maintains equality

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

Tip: Always undo addition/subtraction BEFORE multiplication/division!
Tip: Think of the variable as being "wrapped" - unwrap in reverse order!
2 Subtraction and multiplication
Exercise 2
Solve for x:
\(2x - 7 = 13\)
Definition:

Two-step equation with subtraction: The variable is first multiplied, then a constant is subtracted

Reverse order: Undo subtraction first, then undo multiplication

Original
\(2x - 7 = 13\)
Add 7
\(2x = 20\)
Divide by 2
\(x = 10\)
Step 1: Write the equation

\(2x - 7 = 13\)

Step 2: Undo subtraction by adding 7 to both sides

\(2x - 7 + 7 = 13 + 7\)

\(2x = 20\)

Step 3: Undo multiplication by dividing both sides by 2

\(\frac{2x}{2} = \frac{20}{2}\)

\(x = 10\)

Step 4: Verify the solution

Substitute \(x = 10\) into original: \(2(10) - 7 = 20 - 7 = 13\) ✓

\(x = 10\)
Final answer:

The solution is \(x = 10\)

Applied rules:

• Addition/Subtraction Property: Adding or subtracting the same number from both sides maintains equality

• Multiplication/Division Property: Dividing both sides by the same non-zero number maintains equality

• Reverse Order: Undo operations in reverse of how they were applied

Tip: Look at the variable side: \(2x - 7\) means "multiply by 2, then subtract 7" - so undo "subtract 7" first, then "multiply by 2".
3 Addition and division
Exercise 3
Solve for x:
\(\frac{x}{4} + 3 = 8\)
Definition:

Two-step equation with division: The variable is first divided, then a constant is added

Reverse order: Undo addition first, then undo division

Original
\(\frac{x}{4} + 3 = 8\)
Subtract 3
\(\frac{x}{4} = 5\)
Multiply by 4
\(x = 20\)
Step 1: Write the equation

\(\frac{x}{4} + 3 = 8\)

Step 2: Undo addition by subtracting 3 from both sides

\(\frac{x}{4} + 3 - 3 = 8 - 3\)

\(\frac{x}{4} = 5\)

Step 3: Undo division by multiplying both sides by 4

\(\frac{x}{4} \times 4 = 5 \times 4\)

\(x = 20\)

Step 4: Verify the solution

Substitute \(x = 20\) into original: \(\frac{20}{4} + 3 = 5 + 3 = 8\) ✓

\(x = 20\)
Final answer:

The solution is \(x = 20\)

Applied rules:

• Addition/Subtraction Property: Adding or subtracting the same number from both sides maintains equality

• Multiplication/Division Property: Multiplying both sides by the same non-zero number maintains equality

• Reverse Order: Undo operations in reverse of how they were applied

Tip: When undoing division, multiply by the denominator: \(\frac{x}{4} \times 4 = x\)
Rules and methods, laws,...
\(ax + b = c \Rightarrow x = \frac{c - b}{a}\)
General Form
\(ax - b = c \Rightarrow x = \frac{c + b}{a}\)
Subtraction Form
\(\frac{x}{a} + b = c \Rightarrow x = a(c - b)\)
Division Form
\(\frac{x}{a} - b = c \Rightarrow x = a(c + b)\)
Division/Subtraction Form
Add/Mult
\(ax + b = c\)
Solve: subtract \(b\), then divide by \(a\)
Sub/Mult
\(ax - b = c\)
Solve: add \(b\), then divide by \(a\)
Add/Div
\(\frac{x}{a} + b = c\)
Solve: subtract \(b\), then multiply by \(a\)
Sub/Div
\(\frac{x}{a} - b = c\)
Solve: add \(b\), then multiply by \(a\)
Equality Property: Both sides of an equation must remain equal when performing operations.
Reverse Order: Undo operations in reverse order of how they were applied to the variable.
Solution: Exercises 4 to 5
4 Subtraction and division
Exercise 4
Solve for x:
\(\frac{x}{5} - 2 = 6\)
Definition:

Two-step equation with division and subtraction: The variable is first divided, then a constant is subtracted

Reverse order: Undo subtraction first, then undo division

Original
\(\frac{x}{5} - 2 = 6\)
Add 2
\(\frac{x}{5} = 8\)
Multiply by 5
\(x = 40\)
Step 1: Write the equation

\(\frac{x}{5} - 2 = 6\)

Step 2: Undo subtraction by adding 2 to both sides

\(\frac{x}{5} - 2 + 2 = 6 + 2\)

\(\frac{x}{5} = 8\)

Step 3: Undo division by multiplying both sides by 5

\(\frac{x}{5} \times 5 = 8 \times 5\)

\(x = 40\)

Step 4: Verify the solution

Substitute \(x = 40\) into original: \(\frac{40}{5} - 2 = 8 - 2 = 6\) ✓

\(x = 40\)
Final answer:

The solution is \(x = 40\)

Applied rules:

• Addition/Subtraction Property: Adding or subtracting the same number from both sides maintains equality

• Multiplication/Division Property: Multiplying both sides by the same non-zero number maintains equality

• Reverse Order: Undo operations in reverse of how they were applied

Tip: When undoing division, multiply by the denominator: \(\frac{x}{5} \times 5 = x\)
5 Complex two-step equation
Exercise 5
Solve for x:
\(-4x + 9 = -7\)
Definition:

Complex two-step equation: An equation with negative coefficients and negative results

Same method applies: Undo operations in reverse order, regardless of signs

Original
\(-4x + 9 = -7\)
Subtract 9
\(-4x = -16\)
Divide by -4
\(x = 4\)
Step 1: Write the equation

\(-4x + 9 = -7\)

Step 2: Undo addition by subtracting 9 from both sides

\(-4x + 9 - 9 = -7 - 9\)

\(-4x = -16\)

Step 3: Undo multiplication by dividing both sides by -4

\(\frac{-4x}{-4} = \frac{-16}{-4}\)

\(x = 4\)

Step 4: Verify the solution

Substitute \(x = 4\) into original: \(-4(4) + 9 = -16 + 9 = -7\) ✓

\(x = 4\)
Final answer:

The solution is \(x = 4\)

Applied rules:

• Negative Division: Dividing two negative numbers gives a positive result

• Consistent Method: Same order of operations applies regardless of signs

• Verification: Always substitute back to check accuracy with negative values

Tip: When dividing by a negative number, pay attention to sign changes: \(\frac{-16}{-4} = 4\)
Tip: Negative coefficients don't change the solving method - still undo operations in reverse order!
Comprehensive Guide to Two-Step Equations
\(ax + b = c \Rightarrow x = \frac{c - b}{a}\)
General Solution Formula
Key definitions:

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

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

Order of operations: Operations applied to variables follow PEMDAS/BODMAS rules

Reverse order: To solve, undo operations in reverse of how they were applied

Complete methodology:
  1. Identify operations: Determine what operations are applied to the variable and in what order
  2. Undo in reverse order: Start with the last operation applied and work backwards
  3. Apply inverse operations: Use the opposite operation to both sides of the equation
  4. Isolate the variable: Simplify to get the variable alone on one side
  5. Verify the solution: Substitute the answer back into the original equation
Tip 1: Think of the variable as being "wrapped" in operations - unwrap in reverse order!
Tip 2: Always perform the same operation on both sides to maintain equality.
Tip 3: Check your work by substituting the solution back into the original equation.
Tip 4: Pay special attention to signs when working with negative numbers.
Common errors: Performing operations in the wrong order, forgetting to apply operations to both sides, making sign errors, not verifying solutions.
Exam preparation: Practice all four forms of two-step equations, focus on problems with negative coefficients, master verification techniques.
Formulas to know by heart:

• Addition/Multiplication: \(ax + b = c \Rightarrow x = \frac{c - b}{a}\)

• Subtraction/Multiplication: \(ax - b = c \Rightarrow x = \frac{c + b}{a}\)

• Addition/Division: \(\frac{x}{a} + b = c \Rightarrow x = a(c - b)\)

• Subtraction/Division: \(\frac{x}{a} - b = c \Rightarrow x = a(c + b)\)

Two-Step Equations Workflow

📊
Solving Process
1
Identify Operations
2
Undo Last Operation
3
Undo First Operation
4
Verify
Two-Step Equation Forms
Form: ax + b = c → subtract b, then divide by a
Form: ax - b = c → add b, then divide by a
Form: x/a + b = c → subtract b, then multiply by a
Form: x/a - b = c → add b, then multiply by a
Master These Patterns to Excel in Algebra!

Questions & Answers

Question: Why do we have to undo operations in reverse order? Can't I just do them in the order they appear?

Answer: Great question! Think of it like taking off clothes - if you put on socks first, then shoes, you have to remove shoes first, then socks. With equations, if the variable was multiplied first, then added to, you must subtract first, then divide.

For example, in \(3x + 5 = 20\):

  • The variable \(x\) was first multiplied by 3, then 5 was added
  • To solve, we must do the opposite in reverse order: subtract 5, then divide by 3
  • This "unwraps" the operations in the correct sequence

This follows the principle of reversing the order of operations (PEMDAS/BODMAS backwards).

Question: I sometimes get confused with negative numbers. How do I handle them in two-step equations?

Answer: Negative numbers follow the same solving pattern, but you must be careful with signs:

  • Adding a negative: \(x + (-3) = 5\) is the same as \(x - 3 = 5\), so add 3 to both sides
  • Subtracting a negative: \(x - (-3) = 5\) is the same as \(x + 3 = 5\), so subtract 3 from both sides
  • Multiplying/dividing by a negative: The rules for signs still apply: \(\frac{-12}{-3} = 4\)

Example: In \(-2x + 5 = 11\), subtract 5 from both sides: \(-2x = 6\), then divide by -2: \(x = -3\).

Always verify by substituting back: \(-2(-3) + 5 = 6 + 5 = 11\) ✓

Question: How can I check if I've solved a two-step equation correctly?

Answer: Verification is crucial! Simply substitute your solution back into the original equation:

For \(3x + 5 = 20\) with solution \(x = 5\):

  • Substitute: \(3(5) + 5\)
  • Calculate: \(15 + 5 = 20\)
  • Compare: Does this equal the right side of the original equation? Yes, so the solution is correct!

This verification step catches most errors and builds confidence in your answers.

If your verification doesn't work, retrace your steps - you likely made an error in arithmetic or in applying the inverse operations.