Solved Exercises on Algebraic Equations Basics in Pre-algebra

Master algebraic equations basics: understanding how to solve simple equations and find unknown values through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Addition equation
Exercise 1
Solve: \(x + 5 = 12\)
Definition:

Algebraic equation: A mathematical statement that two expressions are equal, containing at least one variable whose value needs to be found.

Solving method:
  1. Isolate the variable by performing inverse operations
  2. Apply the same operation to both sides of the equation
  3. Simplify both sides
  4. Check the solution by substituting back
Original
\(x + 5 = 12\)
Subtract 5
\(x = 7\)
Check
\(7 + 5 = 12\) ✓
Step 1: Identify the operation

The variable \(x\) is added to 5, so we use the inverse operation (subtraction)

Step 2: Subtract 5 from both sides

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

\(x = 7\)

Step 3: Verify the solution

Substitute \(x = 7\) back into the original equation: \(7 + 5 = 12\) ✓

\(x = 7\)
Final answer:

\(x = 7\)

Applied rules:

• Inverse operations: Addition and subtraction are inverses

• Balancing: Perform the same operation on both sides

• Verification: Always check your solution

2 Subtraction equation
Exercise 2
Solve: \(y - 8 = 15\)
Definition:

Inverse operation: The operation that undoes another operation. For subtraction, the inverse is addition.

Original
\(y - 8 = 15\)
Add 8
\(y = 23\)
Check
\(23 - 8 = 15\) ✓
Step 1: Identify the operation

The variable \(y\) is subtracted by 8, so we use the inverse operation (addition)

Step 2: Add 8 to both sides

\(y - 8 + 8 = 15 + 8\)

\(y = 23\)

Step 3: Verify the solution

Substitute \(y = 23\) back into the original equation: \(23 - 8 = 15\) ✓

\(y = 23\)
Final answer:

\(y = 23\)

Applied rules:

• Inverse operations: Subtraction and addition are inverses

• Balancing: Perform the same operation on both sides

• Verification: Always check your solution

3 Multiplication equation
Exercise 3
Solve: \(3z = 21\)
Definition:

Multiplication equation: An equation where the variable is multiplied by a number. The inverse operation is division.

Original
\(3z = 21\)
Divide by 3
\(z = 7\)
Check
\(3 \times 7 = 21\) ✓
Step 1: Identify the operation

The variable \(z\) is multiplied by 3, so we use the inverse operation (division)

Step 2: Divide both sides by 3

\(\frac{3z}{3} = \frac{21}{3}\)

\(z = 7\)

Step 3: Verify the solution

Substitute \(z = 7\) back into the original equation: \(3 \times 7 = 21\) ✓

\(z = 7\)
Final answer:

\(z = 7\)

Applied rules:

• Inverse operations: Multiplication and division are inverses

• Balancing: Perform the same operation on both sides

• Verification: Always check your solution

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

Algebraic Equation: A mathematical statement showing that two expressions are equal, containing one or more variables.

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

Inverse Operation: An operation that undoes another operation (addition undoes subtraction, multiplication undoes division).

Balancing: Performing the same operation on both sides of an equation to maintain equality.

Verification: Checking that the solution makes the original equation true.

Complete methodology:
  1. Identify the variable: Determine which letter represents the unknown
  2. Identify the operation: Determine what operation is being performed on the variable
  3. Apply inverse operation: Use the inverse 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 the solution by substituting back into the original equation
Tip 1: Always perform the same operation on both sides of the equation
Tip 2: Use inverse operations to isolate the variable
Tip 3: Always verify your solution by substituting back
Tip 4: Keep the variable on the left side when possible for clarity
Common errors: Forgetting to apply operations to both sides, using the wrong inverse operation, not verifying the solution.
Memory aids: "What you do to one side, you do to the other", "Use inverse operations to get the variable alone".
Solution: Exercises 4 to 5
4 Division equation
Exercise 4
Solve: \(\frac{w}{4} = 9\)
Definition:

Division equation: An equation where the variable is divided by a number. The inverse operation is multiplication.

Original
\(\frac{w}{4} = 9\)
Multiply by 4
\(w = 36\)
Check
\(\frac{36}{4} = 9\) ✓
Step 1: Identify the operation

The variable \(w\) is divided by 4, so we use the inverse operation (multiplication)

Step 2: Multiply both sides by 4

\(\frac{w}{4} \times 4 = 9 \times 4\)

\(w = 36\)

Step 3: Verify the solution

Substitute \(w = 36\) back into the original equation: \(\frac{36}{4} = 9\) ✓

\(w = 36\)
Final answer:

\(w = 36\)

Applied rules:

• Inverse operations: Division and multiplication are inverses

• Balancing: Perform the same operation on both sides

• Verification: Always check your solution

5 Two-step equation
Exercise 5
Solve: \(2x + 3 = 11\)
Definition:

Two-step equation: An equation requiring two operations to isolate the variable, typically undoing addition/subtraction first, then multiplication/division.

Original
\(2x + 3 = 11\)
Subtract 3
\(2x = 8\)
Divide by 2
\(x = 4\)
Step 1: Undo addition first

Subtract 3 from both sides: \(2x + 3 - 3 = 11 - 3\)

\(2x = 8\)

Step 2: Undo multiplication

Divide both sides by 2: \(\frac{2x}{2} = \frac{8}{2}\)

\(x = 4\)

Step 3: Verify the solution

Substitute \(x = 4\) back into the original equation: \(2(4) + 3 = 8 + 3 = 11\) ✓

\(x = 4\)
Final answer:

\(x = 4\)

Applied rules:

• Order of operations: Undo operations in reverse order

• Balancing: Perform the same operation on both sides

• Verification: Always check your solution

Comprehensive Summary: Algebraic Equations Basics
\(ax + b = c\)
General Linear Equation
Key definitions:

Algebraic Equation: A mathematical statement that shows two expressions are equal, containing one or more variables whose values must be determined. Example: \(x + 5 = 12\).

Solution: The value of the variable that makes the equation true. For \(x + 5 = 12\), the solution is \(x = 7\).

Inverse Operations: Operations that undo each other: addition and subtraction are inverses, multiplication and division are inverses.

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

Balancing: Maintaining equality by performing the same operation on both sides of the equation.

Complete methodology:
  1. Examine the equation: Identify the variable and operations affecting it
  2. Plan the solution: Determine which inverse operations to apply and in what order
  3. Apply inverse operations: Perform operations to isolate the variable
  4. Maintain balance: Apply the same operation to both sides of the equation
  5. Simplify: Calculate the result after each step
  6. Verify: Check the solution by substituting back into the original equation
Tip 1: Always work to get the variable by itself on one side of the equation
Tip 2: Undo operations in reverse order of operations (PEMDAS backwards)
Tip 3: When solving two-step equations, undo addition/subtraction first, then multiplication/division
Tip 4: Always check your answer by substituting back into the original equation
Tip 5: Keep your work organized with each step clearly shown
Common errors: Forgetting to apply operations to both sides, using the wrong inverse operation, not checking solutions, making calculation errors.
Memory aids: "Do the same thing to both sides", "Undo operations in reverse order", "What you do to one side, you do to the other".
Essential rules to remember:

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

• Inverse operations: Use addition to undo subtraction, subtraction to undo addition, multiplication to undo division, division to undo multiplication

• Isolation principle: Get the variable alone on one side of the equation

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

• Order consideration: For multi-step equations, undo operations in reverse order of operations

Visualization: Equation Balance Concept
Exercise 6: Solution Verification
Visualizing how equations maintain balance:
\(f_1(x) = x + 5\) and \(f_2(x) = 12\) intersect at \(x = 7\)
\(f_3(x) = 3x\) and \(f_4(x) = 21\) intersect at \(x = 7\)
\(f_5(x) = 2x + 3\) and \(f_6(x) = 11\) intersect at \(x = 4\)

Analysis: The chart shows how equations represent balanced relationships.

  • Equations are like balanced scales
  • Solutions occur where functions intersect
  • Operations maintain the balance
  • Verification confirms the solution

Questions & Answers

Question: I don't understand why I need to do the same thing to both sides of the equation. Why can't I just move numbers around?

Answer: Think of an equation like a balanced scale. If you have 5 apples on one side and 5 apples on the other, the scale is balanced.

If you add 2 apples to only one side, the scale becomes unbalanced. To keep it balanced, you must add 2 apples to both sides.

An equation works the same way:

  • \(x + 3 = 7\) means both sides are equal (balanced)
  • If you subtract 3 from only the left side: \(x = 7\), this is wrong!
  • You must subtract 3 from both sides: \(x + 3 - 3 = 7 - 3\), so \(x = 4\)

The equals sign (=) means "these two sides are the same." To keep them the same, you must do the same thing to both sides.

This is called the "balance property" of equations, and it's the foundation of all algebraic solving methods.

Question: How do I know which operation to use first when solving equations? Like in \(2x + 3 = 11\), why do I subtract 3 first instead of dividing by 2?

Answer: You work backwards through the order of operations (PEMDAS/BODMAS). Think about what was done to the variable:

In \(2x + 3 = 11\), to get to the right side, the variable \(x\) was:

  1. Multiplied by 2
  2. Then added 3

To solve, you undo these operations in reverse order:

  1. First, undo the addition (subtract 3)
  2. Then, undo the multiplication (divide by 2)

So: \(2x + 3 = 11\)

  • Subtract 3 from both sides: \(2x = 8\)
  • Divide both sides by 2: \(x = 4\)

This is sometimes called "undoing operations in reverse order" or "working backwards through the order of operations."

Question: Why do I need to check my answer? Isn't solving the equation enough?

Answer: Checking your answer is crucial for several reasons:

Verification: It confirms that your solution is correct. If you substitute your answer back into the original equation and both sides aren't equal, you made a mistake.

Error detection: It helps catch common mistakes like:

  • Sign errors (forgetting negative signs)
  • Calculation mistakes
  • Using the wrong inverse operation
  • Not applying operations to both sides

Confidence building: When you check and find that both sides are equal, you know your answer is correct.

Example: For \(x + 5 = 12\), if you think \(x = 8\):

  • Check: \(8 + 5 = 13\), but the right side is 12
  • Since \(13 \neq 12\), \(x = 8\) is wrong
  • The correct answer is \(x = 7\) because \(7 + 5 = 12\) ✓

Checking takes only a moment but saves you from accepting incorrect answers!