Solved Exercises on One-Step Equations in Integrated Math 1

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

Solution: Exercises 1 to 3
1 Addition equation
Exercise 1
Solve for x:
\(x + 7 = 15\)
Definition:

One-step equation: An equation requiring only one operation to isolate the variable

Addition equation: An equation where the variable is added to a number

Inverse operations method:

To solve addition equations, use the inverse operation: subtraction

  1. Identify the operation on the variable side (addition)
  2. Apply the inverse operation to both sides (subtract the same number)
  3. Isolate the variable
  4. Verify the solution by substituting back
Original
\(x + 7 = 15\)
Subtract 7
\(x = 8\)
Step 1: Write the equation

\(x + 7 = 15\)

Step 2: Subtract 7 from both sides

\(x + 7 - 7 = 15 - 7\)

Step 3: Simplify both sides

\(x = 8\)

Step 4: Verify the solution

Substitute \(x = 8\) into original: \(8 + 7 = 15\) ✓

\(x = 8\)
Final answer:

The solution is \(x = 8\)

Applied rules:

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

• Inverse Operations: Subtraction undoes addition

• Verification: Always substitute back to check accuracy

Tip: Remember: what you do to one side, you must do to the other side!
2 Subtraction equation
Exercise 2
Solve for x:
\(x - 12 = 5\)
Definition:

Subtraction equation: An equation where the variable is subtracted by a number

Inverse operation: Addition undoes subtraction

Original
\(x - 12 = 5\)
Add 12
\(x = 17\)
Step 1: Write the equation

\(x - 12 = 5\)

Step 2: Add 12 to both sides

\(x - 12 + 12 = 5 + 12\)

Step 3: Simplify both sides

\(x = 17\)

Step 4: Verify the solution

Substitute \(x = 17\) into original: \(17 - 12 = 5\) ✓

\(x = 17\)
Final answer:

The solution is \(x = 17\)

Applied rules:

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

• Inverse Operations: Addition undoes subtraction

• Balancing: Whatever operation performed on one side must be done to the other

Tip: Think of the equation as a balance scale - keep it balanced by doing the same thing to both sides!
3 Multiplication equation
Exercise 3
Solve for x:
\(4x = 28\)
Definition:

Multiplication equation: An equation where the variable is multiplied by a number

Inverse operation: Division undoes multiplication

Original
\(4x = 28\)
Divide by 4
\(x = 7\)
Step 1: Write the equation

\(4x = 28\)

Step 2: Divide both sides by 4

\(\frac{4x}{4} = \frac{28}{4}\)

Step 3: Simplify both sides

\(x = 7\)

Step 4: Verify the solution

Substitute \(x = 7\) into original: \(4(7) = 28\) ✓

\(x = 7\)
Final answer:

The solution is \(x = 7\)

Applied rules:

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

• Inverse Operations: Division undoes multiplication

• Fraction Simplification: \(\frac{4x}{4} = x\) and \(\frac{28}{4} = 7\)

Tip: When dividing, write it as a fraction to make cancellation clearer: \(\frac{4x}{4} = x\)
Rules and methods, laws,...
\(x + a = b \Rightarrow x = b - a\)
Addition Equations
\(x - a = b \Rightarrow x = b + a\)
Subtraction Equations
\(ax = b \Rightarrow x = \frac{b}{a}\)
Multiplication Equations
\(\frac{x}{a} = b \Rightarrow x = ab\)
Division Equations
Addition
\(x + a = b\)
Solve by subtracting \(a\) from both sides
Subtraction
\(x - a = b\)
Solve by adding \(a\) to both sides
Multiplication
\(ax = b\)
Solve by dividing both sides by \(a\)
Division
\(\frac{x}{a} = b\)
Solve by multiplying both sides by \(a\)
Equality Property: Both sides of an equation must remain equal when performing operations.
Inverse Operations: Addition and subtraction are inverses; multiplication and division are inverses.
Solution: Exercises 4 to 5
4 Division equation
Exercise 4
Solve for x:
\(\frac{x}{3} = 9\)
Definition:

Division equation: An equation where the variable is divided by a number

Inverse operation: Multiplication undoes division

Original
\(\frac{x}{3} = 9\)
Multiply by 3
\(x = 27\)
Step 1: Write the equation

\(\frac{x}{3} = 9\)

Step 2: Multiply both sides by 3

\(\frac{x}{3} \times 3 = 9 \times 3\)

Step 3: Simplify both sides

\(x = 27\)

Step 4: Verify the solution

Substitute \(x = 27\) into original: \(\frac{27}{3} = 9\) ✓

\(x = 27\)
Final answer:

The solution is \(x = 27\)

Applied rules:

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

• Inverse Operations: Multiplication undoes division

• Fraction Simplification: \(\frac{x}{3} \times 3 = x\) and \(9 \times 3 = 27\)

Tip: Remember: \(\frac{x}{a} \times a = x\) because the denominator cancels out!
5 Fraction coefficient equation
Exercise 5
Solve for x:
\(\frac{2}{3}x = 8\)
Definition:

Fraction coefficient equation: An equation where the variable has a fractional coefficient

Inverse operation: Multiply by the reciprocal to eliminate the fraction

Original
\(\frac{2}{3}x = 8\)
Multiply by reciprocal
\(x = 12\)
Step 1: Write the equation

\(\frac{2}{3}x = 8\)

Step 2: Multiply both sides by the reciprocal of \(\frac{2}{3}\)

The reciprocal of \(\frac{2}{3}\) is \(\frac{3}{2}\)

\(\frac{3}{2} \times \frac{2}{3}x = \frac{3}{2} \times 8\)

Step 3: Simplify both sides

Left side: \(\frac{3}{2} \times \frac{2}{3}x = \frac{6}{6}x = x\)

Right side: \(\frac{3}{2} \times 8 = \frac{24}{2} = 12\)

So: \(x = 12\)

Step 4: Verify the solution

Substitute \(x = 12\) into original: \(\frac{2}{3} \times 12 = \frac{24}{3} = 8\) ✓

\(x = 12\)
Final answer:

The solution is \(x = 12\)

Applied rules:

• Reciprocal Multiplication: To eliminate a fraction coefficient, multiply by its reciprocal

• Fraction Multiplication: \(\frac{a}{b} \times \frac{c}{d} = \frac{ac}{bd}\)

• Reciprocal Property: \(\frac{a}{b} \times \frac{b}{a} = 1\), so \(\frac{a}{b} \times \frac{b}{a}x = x\)

Tip: To find the reciprocal of a fraction, flip the numerator and denominator: \(\frac{2}{3} \rightarrow \frac{3}{2}\)
Tip: Always check your answer by substituting back into the original equation!
Comprehensive Guide to One-Step Equations
\(x + a = b \Rightarrow x = b - a\)
Addition Equations
Key definitions:

One-step equation: An equation that can be solved in a single step by using inverse operations

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

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

Verification: Checking that the solution satisfies the original equation

Complete methodology:
  1. Identify the operation: Determine whether the variable is being added to, subtracted from, multiplied by, or divided by a number
  2. Apply the inverse operation: Perform the opposite operation to both sides of the equation
  3. Isolate the variable: Simplify to get the variable alone on one side
  4. Verify the solution: Substitute the answer back into the original equation
Tip 1: Think of an equation as a balance scale - whatever you do to one side, you must do to the other to keep it balanced.
Tip 2: When dealing with fractions, multiply both sides by the reciprocal of the coefficient.
Tip 3: Always verify your answer by substituting it back into the original equation.
Tip 4: Negative coefficients work the same way - just apply the inverse operation carefully.
Common errors: Forgetting to perform the same operation on both sides, incorrectly identifying inverse operations, making calculation errors with negatives.
Exam preparation: Practice all four types of equations (addition, subtraction, multiplication, division), focus on fraction equations, master verification techniques.
Formulas to know by heart:

• Addition: \(x + a = b \Rightarrow x = b - a\)

• Subtraction: \(x - a = b \Rightarrow x = b + a\)

• Multiplication: \(ax = b \Rightarrow x = \frac{b}{a}\)

• Division: \(\frac{x}{a} = b \Rightarrow x = ab\)

• Fraction coefficient: \(\frac{a}{b}x = c \Rightarrow x = c \times \frac{b}{a}\)

One-Step Equations Workflow

📊
Solving Process
1
Identify Operation
2
Apply Inverse
3
Simplify
4
Verify
Equation Types Visualization
Addition: x + a = b → subtract a
Subtraction: x - a = b → add a
Multiplication: ax = b → divide by a
Division: x/a = b → multiply by a
Master These 4 Types to Excel in Algebra!

Questions & Answers

Question: I'm confused about why we need to do the same operation to both sides of an equation. Why can't we just move numbers around?

Answer: Great question! Think of an equation as a balance scale that must stay balanced. If you have 5 apples on each side of the scale, it's balanced. If you add 2 apples to the left side, you must also add 2 apples to the right side to keep it balanced.

In mathematical terms: If \(x + 3 = 7\), then \(x + 3\) and \(7\) are equal. If you subtract 3 from the left side (\(x + 3 - 3\)), you must also subtract 3 from the right side (\(7 - 3\)) to maintain equality.

This principle ensures that the equation remains true throughout the solving process. Without maintaining balance, you'd get incorrect solutions!

Question: How do I know which inverse operation to use? I sometimes get confused between addition/subtraction and multiplication/division.

Answer: The key is to look at what's happening to your variable:

  • If the variable is being added to a number (like \(x + 5 = 10\)), use subtraction to undo it
  • If the variable is being subtracted by a number (like \(x - 3 = 7\)), use addition to undo it
  • If the variable is being multiplied by a number (like \(4x = 12\)), use division to undo it
  • If the variable is being divided by a number (like \(\frac{x}{2} = 6\)), use multiplication to undo it

Think: "What operation is being performed on the variable? Then do the opposite!"

Example: In \(3x = 15\), the variable \(x\) is multiplied by 3, so divide both sides by 3 to undo it.

Question: I struggle with fraction equations like \(\frac{2}{3}x = 6\). Can you explain why we multiply by the reciprocal?

Answer: When you have a fraction coefficient like \(\frac{2}{3}\) multiplying your variable, you want to turn that coefficient into 1 (so you have \(1x\) which is just \(x\)).

The reciprocal of \(\frac{2}{3}\) is \(\frac{3}{2}\) because \(\frac{2}{3} \times \frac{3}{2} = \frac{6}{6} = 1\).

So in \(\frac{2}{3}x = 6\), multiplying both sides by \(\frac{3}{2}\):

  • Left side: \(\frac{3}{2} \times \frac{2}{3}x = 1x = x\)
  • Right side: \(\frac{3}{2} \times 6 = \frac{18}{2} = 9\)

Therefore: \(x = 9\)

The reciprocal effectively "cancels out" the fraction coefficient!