📐 Concept diagram

01-02 - Linear Equations

Phase: 1 | Subject: 01-02 Prerequisites: 01-01-algebraic-expressions.md (substitution, like terms, expanding brackets) Next subject: 01-03-linear-inequalities.md

Learning Objectives

By the end of this subject, you will be able to:

1. Solve one-step and two-step linear equations using inverse operations
2. Solve equations containing brackets and variables on both sides
3. Solve fractional linear equations by clearing denominators
4. Verify solutions by substituting back into the original equation
5. Recognise when an equation has no solution or infinitely many solutions

Core Content

What is a Linear Equation?

A linear equation is an equation where the highest power of the variable is 1. It forms a straight line when graphed.

Examples of linear equations: - $2x + 5 = 13$ (one variable) - $3y - 7 = 2y + 4$ (variables on both sides) - $(x + 3)/4 = 5$ (fractional)

Not linear: - $x^2 + 3 = 7$ (quadratic - power of 2) - $2^x = 8$ (exponential - variable in exponent)

The Balance Method

Think of an equation as a balanced scale. Whatever you do to one side, you must do to the other to keep it balanced.

Inverse operations undo each other: - Addition ↔ Subtraction - Multiplication ↔ Division

Solving One-Step Equations

Example: $x + 7 = 15$

1. Identify what's being done to x: adding 7
2. Undo it: subtract 7 from both sides
3. $x + 7 - 7 = 15 - 7$
4. $x = 8$

Check: $8 + 7 = 15$ ✓

Solving Two-Step Equations

Example: $3x - 4 = 11$

1. Undo subtraction first: $3x - 4 + 4 = 11 + 4$
2. $3x = 15$
3. Undo multiplication: $3x / 3 = 15 / 3$
4. $x = 5$

Check: $3(5) - 4 = 15 - 4 = 11$ ✓

Equations with Brackets

Use the distributive law to expand brackets first.

Example: $2(3x - 5) = 22$

1. Expand: $6x - 10 = 22$
2. Add 10: $6x = 32$
3. Divide by 6: $x = 32/6 = 16/3$

Check: $2(3(16/3) - 5) = 2(16 - 5) = 2(11) = 22$ ✓

Variables on Both Sides

Example: $4x + 7 = 2x + 15$

1. Move x terms to left: $4x - 2x + 7 = 15$
2. Simplify: $2x + 7 = 15$
3. Subtract 7: $2x = 8$
4. Divide by 2: $x = 4$

Check: $4(4) + 7 = 16 + 7 = 23$ and $2(4) + 15 = 8 + 15 = 23$ ✓

Fractional Equations

Example: $(2x + 1)/3 = 7$

1. Multiply both sides by 3: $2x + 1 = 21$
2. Subtract 1: $2x = 20$
3. Divide by 2: $x = 10$

Example with multiple fractions: $(x - 2)/4 + (x + 1)/2 = 3$

1. Find LCD: 4
2. Multiply all terms by 4: $(x - 2) + 2(x + 1) = 12$
3. Expand: $x - 2 + 2x + 2 = 12$
4. Simplify: $3x = 12$
5. $x = 4$

Special Cases

No solution: $2x + 3 = 2x + 7$ Subtract 2x: $3 = 7$ (FALSE) This equation has no solution.

Infinitely many solutions: $3(x - 2) = 3x - 6$ Expand: $3x - 6 = 3x - 6$ Subtract 3x: $-6 = -6$ (TRUE for ALL x) Every real number is a solution.

Key Terms

• Inverse operations

Worked Examples

Example 1: Solve 5y - 8 = 2y + 7

1. Move y terms to left: $5y - 2y - 8 = 7$
2. Simplify: $3y - 8 = 7$
3. Add 8: $3y = 15$
4. Divide by 3: $y = 5$

Check: $5(5) - 8 = 25 - 8 = 17$ and $2(5) + 7 = 10 + 7 = 17$ ✓

Example 2: Solve 4(2z + 1) = 3(z - 2) + 11

1. Expand left: $8z + 4$
2. Expand right: $3z - 6 + 11 = 3z + 5$
3. Equation: $8z + 4 = 3z + 5$
4. Move z terms: $8z - 3z + 4 = 5$
5. $5z + 4 = 5$
6. Subtract 4: $5z = 1$
7. $z = 1/5 = 0.2$

Check: $4(2(0.2) + 1) = 4(0.4 + 1) = 4(1.4) = 5.6$ and $3(0.2 - 2) + 11 = 3(-1.8) + 11 = -5.4 + 11 = 5.6$ ✓

Example 3: Solve (3w + 2)/5 = (w - 1)/3

1. Cross multiply: $3(3w + 2) = 5(w - 1)$
2. Expand: $9w + 6 = 5w - 5$
3. Move w terms: $9w - 5w + 6 = -5$
4. $4w + 6 = -5$
5. Subtract 6: $4w = -11$
6. $w = -11/4 = -2.75$

Check: $(3(-2.75) + 2)/5 = (-8.25 + 2)/5 = -6.25/5 = -1.25$ and $(-2.75 - 1)/3 = -3.75/3 = -1.25$ ✓

Quiz

Q1: What does the concept of Inverse operations primarily refer to in this subject?

A) The definition and application of Inverse operations B) A historical anecdote about Inverse operations C) A visual representation of Inverse operations D) A computational error related to Inverse operations

Correct: A)

• If you chose A: Inverse operations is defined as: the definition and application of inverse operations. The other options describe different aspects that are not the primary focus. Correct!
• If you chose B: This is incorrect. Inverse operations is defined as: the definition and application of inverse operations. The other options describe different aspects that are not the primary focus.
• If you chose C: This is incorrect. Inverse operations is defined as: the definition and application of inverse operations. The other options describe different aspects that are not the primary focus.
• If you chose D: This is incorrect. Inverse operations is defined as: the definition and application of inverse operations. The other options describe different aspects that are not the primary focus.

Q2: What is the primary purpose of What Is A Linear Equation??

A) It is used only in advanced research contexts B) It is primarily a historical notation system C) It is used to what is a linear equation? in mathematical analysis D) It replaces all other methods in this domain

Correct: C)

• If you chose A: This is incorrect. What Is A Linear Equation? serves the purpose described in the correct answer. The other options misrepresent its role.
• If you chose B: This is incorrect. What Is A Linear Equation? serves the purpose described in the correct answer. The other options misrepresent its role.
• If you chose C: What Is A Linear Equation? serves the purpose described in the correct answer. The other options misrepresent its role. Correct!
• If you chose D: This is incorrect. What Is A Linear Equation? serves the purpose described in the correct answer. The other options misrepresent its role.

Q3: Which statement about The Balance Method is TRUE?

A) The Balance Method is not related to this subject B) The Balance Method is an advanced topic beyond this subject's scope C) The Balance Method is mentioned only as a historical footnote D) The Balance Method is a fundamental concept covered in this subject

Correct: D)

• If you chose A: This is incorrect. The Balance Method is a fundamental concept covered in this subject. This subject covers The Balance Method as part of its core content.
• If you chose B: This is incorrect. The Balance Method is a fundamental concept covered in this subject. This subject covers The Balance Method as part of its core content.
• If you chose C: This is incorrect. The Balance Method is a fundamental concept covered in this subject. This subject covers The Balance Method as part of its core content.
• If you chose D: The Balance Method is a fundamental concept covered in this subject. This subject covers The Balance Method as part of its core content. Correct!

Q4: Based on the worked examples in this subject, what is the correct result?

A) An unrelated numerical value B) A different result from a common mistake C) 3` D) The inverse of the correct answer

Correct: C)

• If you chose A: This is incorrect. The worked examples show that the result is 3`. The other options represent common errors.
• If you chose B: This is incorrect. The worked examples show that the result is 3`. The other options represent common errors.
• If you chose C: The worked examples show that the result is 3`. The other options represent common errors. Correct!
• If you chose D: This is incorrect. The worked examples show that the result is 3`. The other options represent common errors.

Q5: How are The Balance Method and Solving One-Step Equations related?

A) The Balance Method and Solving One-Step Equations are completely unrelated topics B) The Balance Method and Solving One-Step Equations are closely related concepts C) The Balance Method is a special case of Solving One-Step Equations D) The Balance Method is the inverse of Solving One-Step Equations

Correct: B)

• If you chose A: This is incorrect. Both The Balance Method and Solving One-Step Equations are covered in this subject as interconnected topics.
• If you chose B: Both The Balance Method and Solving One-Step Equations are covered in this subject as interconnected topics. Correct!
• If you chose C: This is incorrect. Both The Balance Method and Solving One-Step Equations are covered in this subject as interconnected topics.
• If you chose D: This is incorrect. Both The Balance Method and Solving One-Step Equations are covered in this subject as interconnected topics.

Q6: What is a common pitfall when working with Solving Two-Step Equations?

A) Solving Two-Step Equations has no common misconceptions B) A common mistake is confusing Solving Two-Step Equations with a similar concept C) Solving Two-Step Equations is always computed the same way in all contexts D) The main error with Solving Two-Step Equations is using it when it is not needed

Correct: B)

• If you chose A: This is incorrect. Students often confuse Solving Two-Step Equations with similar-sounding or related concepts. Pay attention to the precise definitions.
• If you chose B: Students often confuse Solving Two-Step Equations with similar-sounding or related concepts. Pay attention to the precise definitions. Correct!
• If you chose C: This is incorrect. Students often confuse Solving Two-Step Equations with similar-sounding or related concepts. Pay attention to the precise definitions.
• If you chose D: This is incorrect. Students often confuse Solving Two-Step Equations with similar-sounding or related concepts. Pay attention to the precise definitions.

Q7: When should you apply Equations With Brackets?

A) Equations With Brackets is not practically useful B) Apply Equations With Brackets to solve problems in this subject's domain C) Use Equations With Brackets only in pure mathematics contexts D) Avoid Equations With Brackets unless explicitly instructed

Correct: B)

• If you chose A: This is incorrect. Equations With Brackets is a practical tool used throughout this subject to solve relevant problems.
• If you chose B: Equations With Brackets is a practical tool used throughout this subject to solve relevant problems. Correct!
• If you chose C: This is incorrect. Equations With Brackets is a practical tool used throughout this subject to solve relevant problems.
• If you chose D: This is incorrect. Equations With Brackets is a practical tool used throughout this subject to solve relevant problems.

Practice Problems

1. Solve: $x + 9 = 21$
   Click for answer

$x = 12$

1. Solve: $7 - 2y = 3$
   Click for answer

$2y = 4$, so $y = 2$

1. Solve: $5(2x - 3) = 35$
   Click for answer

$10x - 15 = 35$, $10x = 50$, $x = 5$

1. Solve: $3x + 4 = 2x - 5$
   Click for answer

$x = -9$

1. Solve: $(x + 4)/2 = 6$
   Click for answer

$x + 4 = 12$, $x = 8$

1. Solve: $2(x - 3) + 5 = x + 1$
   Click for answer

$2x - 6 + 5 = x + 1$, $2x - 1 = x + 1$, $x = 2$

1. Solve: $(3y - 1)/4 = (y + 5)/2$
   Click for answer

Cross multiply: $2(3y - 1) = 4(y + 5)$, $6y - 2 = 4y + 20$, $2y = 22$, $y = 11$

Summary

Key takeaways:

• Always perform the same operation on both sides to maintain balance
• Use inverse operations to isolate the variable
• Expand brackets before solving
• For fractional equations, multiply by the LCD to clear denominators
• Always verify your solution by substituting back
• Equations may have no solution or infinitely many solutions

Pitfalls

• Performing operations on only one side of the equation. Every operation must be applied to both sides to maintain balance. Adding 5 to the left side but forgetting the right side is the most fundamental error in equation solving.
• Forgetting to multiply every term by the LCD when clearing fractions. In (x − 2)/4 + (x + 1)/2 = 3, multiply ALL terms by 4: (x − 2) + 2(x + 1) = 12. A term without a denominator (like the 3) must also be multiplied.
• Losing a negative sign when expanding brackets. In 4(2z + 1) = 3(z − 2) + 11, the right side expands to 3z − 6 + 11 = 3z + 5. Missing the −6 produces the wrong equation.
• Failing to check solutions by substitution. Many errors (signs, arithmetic) go undetected because students skip verification. Substitute your answer back into the original equation — it takes seconds and catches most mistakes.
• Missing special cases. An equation like 2x + 3 = 2x + 7 simplifies to 3 = 7, which has NO solution. An equation like 3(x − 2) = 3x − 6 simplifies to −6 = −6, which is true for ALL x. Not every linear equation has exactly one answer.

Next Steps

Next up: 01-03-linear-inequalities.md