Solved Exercises on One-Step Addition & Subtraction Equations in Pre-algebra

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

Solution: Exercises 1 to 3
1 Addition equation
Exercise 1
Solve: \(x + 8 = 15\)
Definition:

Addition equation: An equation where a number is added to the variable. To solve, subtract the same number from both sides to isolate the variable.

Solving method:
  1. Identify the operation (addition in this case)
  2. Apply the inverse operation (subtraction) to both sides
  3. Simplify both sides
  4. Verify the solution
Original
\(x + 8 = 15\)
Subtract 8
\(x = 7\)
Check
\(7 + 8 = 15\) ✓
Step 1: Identify the operation

The variable \(x\) has 8 added to it, so we need to subtract 8 from both sides

Step 2: Apply inverse operation

Subtract 8 from both sides: \(x + 8 - 8 = 15 - 8\)

Step 3: Simplify

Left side: \(x + 8 - 8 = x\)

Right side: \(15 - 8 = 7\)

So: \(x = 7\)

Step 4: Verify the solution

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

\(x = 7\)
Final answer:

\(x = 7\)

Applied rules:

• Inverse operations: Subtraction undoes addition

• Balance principle: Perform the same operation on both sides

• Verification: Always check your solution

2 Subtraction equation
Exercise 2
Solve: \(y - 12 = 4\)
Definition:

Subtraction equation: An equation where a number is subtracted from the variable. To solve, add the same number to both sides to isolate the variable.

Original
\(y - 12 = 4\)
Add 12
\(y = 16\)
Check
\(16 - 12 = 4\) ✓
Step 1: Identify the operation

The variable \(y\) has 12 subtracted from it, so we need to add 12 to both sides

Step 2: Apply inverse operation

Add 12 to both sides: \(y - 12 + 12 = 4 + 12\)

Step 3: Simplify

Left side: \(y - 12 + 12 = y\)

Right side: \(4 + 12 = 16\)

So: \(y = 16\)

Step 4: Verify the solution

Substitute \(y = 16\) back into the original equation: \(16 - 12 = 4\) ✓

\(y = 16\)
Final answer:

\(y = 16\)

Applied rules:

• Inverse operations: Addition undoes subtraction

• Balance principle: Perform the same operation on both sides

• Verification: Always check your solution

3 Addition with negative number
Exercise 3
Solve: \(z + (-6) = 10\)
Definition:

Addition with negative number: Adding a negative number is the same as subtracting the positive number. To solve, add the positive number to both sides.

Original
\(z + (-6) = 10\)
Add 6
\(z = 16\)
Check
\(16 + (-6) = 10\) ✓
Step 1: Understand the operation

\(z + (-6)\) is the same as \(z - 6\), so we need to add 6 to both sides

Step 2: Apply inverse operation

Add 6 to both sides: \(z + (-6) + 6 = 10 + 6\)

Step 3: Simplify

Left side: \(z + (-6) + 6 = z + 0 = z\)

Right side: \(10 + 6 = 16\)

So: \(z = 16\)

Step 4: Verify the solution

Substitute \(z = 16\) back into the original equation: \(16 + (-6) = 10\) ✓

\(z = 16\)
Final answer:

\(z = 16\)

Applied rules:

• Adding negative: \(a + (-b) = a - b\)

• Inverse operations: Add positive number to undo negative addition

• Balance principle: Perform the same operation on both sides

Rules and methods, laws,...
\(x + a = b\)
Addition Equation
Addition
\(x + a = b\)
Solution: \(x = b - a\)
Subtraction
\(x - a = b\)
Solution: \(x = b + a\)
Negative Addition
\(x + (-a) = b\)
Solution: \(x = b + a\)
Negative Subtraction
\(x - (-a) = b\)
Solution: \(x = b - a\)
Key definitions:

One-Step Addition Equation: An equation of the form \(x + a = b\) where \(a\) and \(b\) are constants. The solution is \(x = b - a\).

One-Step Subtraction Equation: An equation of the form \(x - a = b\) where \(a\) and \(b\) are constants. The solution is \(x = b + a\).

Inverse Operations: Operations that undo each other. Addition and subtraction are inverse operations.

Balance Principle: Whatever is done to one side of an equation must be done to the other side to maintain equality.

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 addition or subtraction
  2. Identify the operation: See what operation is being performed on the variable
  3. Apply inverse operation: Use the opposite operation to isolate the variable
  4. Balance both sides: Perform the same 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: Adding a negative is the same as subtracting a positive
Tip 2: Subtracting a negative is the same as adding a positive
Tip 3: Always perform the same operation on both sides
Tip 4: Always verify your solution by substituting back
Common errors: Forgetting to apply operations to both sides, using the wrong inverse operation, not checking the solution, making sign errors with negative numbers.
Memory aids: "What you do to one side, you do to the other", "Addition and subtraction are opposites", "Undo the operation to isolate the variable".
Solution: Exercises 4 to 5
4 Subtraction with negative number
Exercise 4
Solve: \(w - (-3) = 8\)
Definition:

Subtraction with negative number: Subtracting a negative number is the same as adding the positive number. To solve, subtract the positive number from both sides.

Original
\(w - (-3) = 8\)
Subtract 3
\(w = 5\)
Check
\(5 - (-3) = 8\) ✓
Step 1: Understand the operation

\(w - (-3)\) is the same as \(w + 3\), so we need to subtract 3 from both sides

Step 2: Apply inverse operation

Subtract 3 from both sides: \(w - (-3) - 3 = 8 - 3\)

Step 3: Simplify

Left side: \(w - (-3) - 3 = w + 3 - 3 = w\)

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

So: \(w = 5\)

Step 4: Verify the solution

Substitute \(w = 5\) back into the original equation: \(5 - (-3) = 5 + 3 = 8\) ✓

\(w = 5\)
Final answer:

\(w = 5\)

Applied rules:

• Subtracting negative: \(a - (-b) = a + b\)

• Inverse operations: Subtract positive number to undo negative subtraction

• Balance principle: Perform the same operation on both sides

5 Real-world application
Exercise 5
Tom has some money in his wallet. He spends $15 and now has $25 left. How much money did he start with? Write and solve the equation.
Definition:

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

Set up
\(x - 15 = 25\)
Add 15
\(x = 40\)
Check
\(40 - 15 = 25\) ✓
Step 1: Define the variable

Let \(x\) = the amount of money Tom started with

Step 2: Set up the equation

He started with \(x\) dollars, spent 15 dollars, and has 25 dollars left: \(x - 15 = 25\)

Step 3: Solve the equation

Add 15 to both sides: \(x - 15 + 15 = 25 + 15\), so \(x = 40\)

Step 4: Verify the solution

If Tom started with $40 and spent $15, he has \(40 - 15 = 25\) dollars left ✓

\(x = 40\)
Final answer:

Tom started with $40. The equation is \(x - 15 = 25\), and the solution is \(x = 40\).

Applied rules:

• Word problem translation: Convert words into mathematical expressions

• Subtraction equation: Add to both sides to solve

• Verification: Check that solution makes sense in context

Comprehensive Summary: One-Step Addition & Subtraction Equations
\(x + a = b \quad \text{or} \quad x - a = b\)
General Forms
Key definitions:

One-Step Addition Equation: An equation of the form \(x + a = b\) where the variable has a number added to it. The solution is found by subtracting \(a\) from both sides: \(x = b - a\).

One-Step Subtraction Equation: An equation of the form \(x - a = b\) where the variable has a number subtracted from it. The solution is found by adding \(a\) to both sides: \(x = b + a\).

Inverse Operations: Operations that undo each other. Addition and subtraction are inverse operations: adding a number and then subtracting 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 addition (\(+\)) or subtraction (\(-\))
  2. Identify the operation on the variable: See what operation is being performed with the variable
  3. Select the inverse operation: Use subtraction to undo addition, addition to undo subtraction
  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 adding a negative number is the same as subtracting a positive number
Tip 2: Remember that subtracting a negative number is the same as adding a positive number
Tip 3: Always keep the variable on one side and move numbers to the other side
Tip 4: Picture the equation as a balanced scale - both sides must stay equal
Tip 5: Always verify your answer by substituting it back into the original equation
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: "Addition and subtraction 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: Addition and subtraction are inverses of each other

• Solution for addition: For \(x + a = b\), the solution is \(x = b - a\)

• Solution for subtraction: For \(x - a = b\), the solution is \(x = b + a\)

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

Visualization: Addition & Subtraction Relationships
Exercise 6: Equation Visualization
Visualizing how addition and subtraction equations work:
\(f_1(x) = x + 8\) and \(g_1(x) = 15\) intersect at \(x = 7\)
\(f_2(x) = x - 12\) and \(g_2(x) = 4\) intersect at \(x = 16\)
\(f_3(x) = x + (-6)\) and \(g_3(x) = 10\) intersect at \(x = 16\)

Analysis: The chart shows how equations represent balanced relationships.

  • Addition equations shift lines upward
  • Subtraction equations shift lines downward
  • Solutions occur where functions intersect
  • Each type maintains the balance principle

Questions & Answers

Question: I get confused when there are negative numbers. Why does subtracting a negative number become addition?

Answer: Think of subtraction as "adding the opposite." When you subtract a negative number, you're adding the opposite of that negative number, which is positive.

Conceptual explanation:

  • \(a - b\) means "a plus the opposite of b"
  • \(a - (-b)\) means "a plus the opposite of -b"
  • The opposite of \(-b\) is \(+b\), so \(a - (-b) = a + b\)

Real-world analogy: If you owe $5 (that's \(-5\)) and someone forgives your debt (subtracting \(-5\)), you're $5 richer: \(0 - (-5) = 0 + 5 = 5\).

Examples:

  • \(7 - (-3) = 7 + 3 = 10\)
  • \(x - (-5) = x + 5\)
  • \(12 - (-8) = 12 + 8 = 20\)

Remember: "Subtracting a negative is adding a positive."

Question: How do I know whether to add or subtract when solving these equations? It seems confusing.

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

Inverse operations:

  • If the variable has addition, use subtraction to undo it
  • If the variable has subtraction, use addition to undo it

Examples:

  • \(x + 7 = 12\): Variable has addition of 7, so subtract 7 from both sides: \(x = 12 - 7 = 5\)
  • \(x - 4 = 9\): Variable has subtraction of 4, so add 4 to both sides: \(x = 9 + 4 = 13\)
  • \(y + (-3) = 8\): Variable has addition of -3 (same as subtraction of 3), so subtract -3 (same as add 3): \(y = 8 + 3 = 11\)

The key is to ask: "What was done to the variable?" Then do the opposite to both sides.

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 \(x + 5 = 10\), both sides are equal (balanced).

If you subtract 5 from only the left side, you get \(x = 10\), but now the scale is unbalanced because \(x\) doesn't equal 10!

To keep the balance:

  • Left side: \(x + 5 - 5 = x\)
  • Right side: \(10 - 5 = 5\)
  • 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.