Solving Linear Equations

Part 1: Foundations & One-Step Equations 🎯

Welcome to Solving Linear Equations! This is one of the most important skills in all of algebra — and it shows up everywhere from physics to finance.

What is a Linear Equation?

A linear equation is an equation where the variable (usually $x$) has an exponent of 1. No $x^2$, no $\sqrt{x}$, no $\frac{1}{x}$ — just plain $x$.

✅ Linear: $3x + 5 = 14$, $\;\; 2(x - 7) = 10$, $\;\; \frac{x}{4} = 9$

❌ Not Linear: $x^2 + 3 = 12$, $\;\; \sqrt{x} = 5$, $\;\; \frac{3}{x} = 7$

The Golden Rule of Equations:

Whatever you do to one side, you must do to the other side.

This is how we keep the equation balanced — like a scale. If you add 5 to the left, you must add 5 to the right. If you multiply the left by 3, you multiply the right by 3. Always.

Quick Check — Identifying Linear Equations 🔍

Solving by Addition or Subtraction ➕➖

When a number is added to or subtracted from the variable, we use the inverse operation to undo it.

Example 1: Solve $x + 7 = 12$

The $+7$ is added to $x$. To undo it, subtract 7 from both sides:

$x + 7 - 7 = 12 - 7$ $x = 5$

Check: $5 + 7 = 12$ ✓

Example 2: Solve $x - 4 = 9$

The $-4$ is subtracted from $x$. To undo it, add 4 to both sides:

$x - 4 + 4 = 9 + 4$ $x = 13$

Check: $13 - 4 = 9$ ✓

Key Insight: Addition and subtraction are inverse operations — they "undo" each other. We're not just moving numbers around; we're performing the same operation on both sides to maintain balance.

Practice: Addition & Subtraction Equations 🧮

Solve each equation for $x$. Enter just the number (or negative number).

1. $x + 15 = 23$

2. $x - 9 = -3$

3. $x + 2.5 = 7$

Solving by Multiplication or Division ✖️➗

When a variable is multiplied or divided by a number, we again use the inverse operation.

Example 3: Solve $4x = 28$

$x$ is multiplied by 4. To undo it, divide both sides by 4:

$\frac{4x}{4} = \frac{28}{4}$ $x = 7$

Check: $4(7) = 28$ ✓

Example 4: Solve $\frac{x}{3} = -5$

$x$ is divided by 3. To undo it, multiply both sides by 3:

$\frac{x}{3} \cdot 3 = -5 \cdot 3$ $x = -15$

Check: $\frac{-15}{3} = -5$ ✓

Example 5: Solve $-6x = 42$

Same process — divide by $-6$:

$\frac{-6x}{-6} = \frac{42}{-6}$ $x = -7$

Check: $-6(-7) = 42$ ✓

Watch the signs! When you divide or multiply by a negative number, the sign of your answer changes.

Practice: Multiplication & Division Equations 🧮

Solve each equation for $x$.

1. $7x = -49$

2. $\frac{x}{5} = 12$

3. $-3x = 27$

Concept Check — Inverse Operations 🔍

Match each equation with the correct first step to solve it.

Common Mistakes to Avoid ⚠️

Mistake 1: Doing different operations to each side

❌ $x + 5 = 12 \Rightarrow x = 12 + 5 = 17$

✅ $x + 5 = 12 \Rightarrow x = 12 - 5 = 7$

You need to subtract 5 (the inverse), not add it again!

Mistake 2: Forgetting the negative sign

❌ $-4x = 20 \Rightarrow x = 5$

✅ $-4x = 20 \Rightarrow x = \frac{20}{-4} = -5$

When dividing by a negative, the answer is negative!

Mistake 3: Not checking your answer

Always plug your answer back in:

• If $x = -5$: Does $-4(-5) = 20$? → $20 = 20$ ✓
• This takes 5 seconds and catches many errors!

Pro Tip: After solving, ask yourself "Does this make sense?" If $x + 100 = 103$ and you got $x = 203$, something went wrong.

Part 1 Exit Quiz — One-Step Equations ✅

Solve each equation and select the correct answer.

Part 1 Complete! 🎉

You've mastered one-step equations — the building blocks of algebra!

Key Takeaways:

• ✅ Addition undoes subtraction (and vice versa)
• ✅ Multiplication undoes division (and vice versa)
• ✅ Whatever you do to one side, do to the other
• ✅ Watch your negative signs — they're the #1 source of errors
• ✅ Always check your answer by plugging it back in

Next Up: Part 2 — Two-Step Equations

Now we'll combine these operations: equations like $3x + 5 = 20$ require TWO steps to solve. The order matters!

Part 2: Two-Step Equations 🔢

Now we level up! Two-step equations require — you guessed it — two operations to solve.

The Big Idea: Most two-step equations look like this:

$ax + b = c$

where $a$, $b$, and $c$ are numbers. Your job is to find $x$.

The Order of Operations (in Reverse!):

When solving equations, you undo operations in reverse order — the opposite of PEMDAS:

1. First: Undo addition or subtraction (get the term with $x$ alone)
2. Then: Undo multiplication or division (isolate $x$ completely)

Think of it like getting dressed vs. getting undressed. You put your shoes on last, but take them off first! Similarly, the last operation applied to $x$ gets undone first.

Example 1: Solve $3x + 5 = 20$

Step 1: Undo the $+5$ (subtract 5 from both sides)

$3x + 5 - 5 = 20 - 5$ $3x = 15$

Step 2: Undo the $\times 3$ (divide both sides by 3)

$\frac{3x}{3} = \frac{15}{3}$ $x = 5$

Check: $3(5) + 5 = 15 + 5 = 20$ ✓

Example 2: Solve $\frac{x}{4} - 7 = 1$

Step 1: Undo the $-7$ (add 7 to both sides)

$\frac{x}{4} - 7 + 7 = 1 + 7$ $\frac{x}{4} = 8$

Step 2: Undo the $\div 4$ (multiply both sides by 4)

$\frac{x}{4} \cdot 4 = 8 \cdot 4$ $x = 32$

Check: $\frac{32}{4} - 7 = 8 - 7 = 1$ ✓

Concept Check — Order of Operations 🎯

Understanding WHY we undo in reverse order is crucial for harder equations.

Two-Step Equations with Negatives ⚠️

Negative coefficients are where most students make errors. Let's be extra careful here.

Example 3: Solve $-2x + 9 = 3$

Step 1: Subtract 9 from both sides

$-2x + 9 - 9 = 3 - 9$ $-2x = -6$

Step 2: Divide both sides by $-2$

$\frac{-2x}{-2} = \frac{-6}{-2}$ $x = 3$

Check: $-2(3) + 9 = -6 + 9 = 3$ ✓

Example 4: Solve $14 - 5x = -1$

Be careful! This is really $-5x + 14 = -1$.

Step 1: Subtract 14 from both sides

$14 - 5x - 14 = -1 - 14$ $-5x = -15$

Step 2: Divide both sides by $-5$

$x = \frac{-15}{-5} = 3$

Check: $14 - 5(3) = 14 - 15 = -1$ ✓

Remember: $\frac{\text{negative}}{\text{negative}} = \text{positive}$

Practice: Two-Step Equations 🧮

Solve each equation for $x$. Enter just the number.

1. $4x + 3 = 31$

2. $\frac{x}{6} - 2 = 5$

3. $-3x + 10 = -8$

Setting Up Two-Step Equations 📖

Real problems don't come pre-written as equations. You need to translate words into algebra.

Key phrases to watch for:

Example 5: A gym charges a $25 registration fee plus $40 per month. If you've paid $225 total, how many months have you been a member?

Set up: Let $m$ = number of months

$40m + 25 = 225$

Solve: $40m = 200$ $m = 5 \text{ months}$

Check: $40(5) + 25 = 200 + 25 = 225$ ✓

Word Problem Practice 📝

Part 2 Exit Challenge 🏆

Solve each equation. These are a step up from the earlier practice!

1. $-7x - 4 = 31$

2. $\frac{x}{-3} + 8 = 2$

3. $15 - 2x = 27$

Part 2 Complete! 🎉

You can now solve two-step linear equations with confidence!

Key Takeaways:

• ✅ Undo addition/subtraction first, then multiplication/division
• ✅ Think "reverse PEMDAS" — undo operations in reverse order
• ✅ Be extra careful with negative coefficients
• ✅ Translate word problems into equations by identifying the variable and operations
• ✅ Always check by substituting back into the original equation

Next Up: Part 3 — Multi-Step Equations & Variables on Both Sides

Things get more interesting! You'll learn to simplify first, then solve — and handle equations where $x$ shows up on BOTH sides.

Part 3: Multi-Step Equations & Variables on Both Sides ⚡

Now we tackle the equations that look intimidating but follow the same core principles you already know.

What's new?

• Equations that need simplifying first (combining like terms, distributing)
• Equations with the variable on both sides (like $5x + 3 = 2x + 15$)

The General Strategy:

Don't memorize these as rigid rules — understand the goal: get $x$ alone on one side.

Simplify First: Combining Like Terms 🔧

Before solving, simplify each side of the equation separately.

Example 1: Solve $3x + 7 + 2x - 4 = 18$

Step 1: Combine like terms on the left

• $3x + 2x = 5x$ (variable terms)
• $7 - 4 = 3$ (constant terms)

$5x + 3 = 18$

Step 2: Now it's a two-step equation! $5x = 15$ $x = 3$

Check: $3(3) + 7 + 2(3) - 4 = 9 + 7 + 6 - 4 = 18$ ✓

Example 2: Solve $4(x + 3) = 28$

Step 1: Distribute the 4

$4x + 12 = 28$

Step 2: Solve the two-step equation $4x = 16$ $x = 4$

Check: $4(4 + 3) = 4(7) = 28$ ✓

Practice: Distribute & Solve 🧮

Solve each equation for $x$.

1. $2(x - 5) = 14$

2. $-3(x + 4) = 15$

3. $5(2x + 1) - 3 = 32$

Variables on Both Sides 🔄

What if $x$ appears on both sides of the equation? We need to collect all the variable terms on one side.

Example 3: Solve $5x + 3 = 2x + 15$

Step 1: Get all $x$-terms on one side

Subtract $2x$ from both sides: $5x - 2x + 3 = 2x - 2x + 15$ $3x + 3 = 15$

Step 2: Solve the two-step equation $3x = 12$ $x = 4$

Check: Left: $5(4) + 3 = 23$. Right: $2(4) + 15 = 23$ ✓

Example 4: Solve $7x - 2 = 3x + 14$

Step 1: Subtract $3x$ from both sides $4x - 2 = 14$

Step 2: Add 2 to both sides $4x = 16$

Step 3: Divide by 4 $x = 4$

Check: Left: $7(4) - 2 = 26$. Right: $3(4) + 14 = 26$ ✓

Pro Tip: You can move the variable to either side. It often helps to move the smaller $x$-coefficient to the other side, so you avoid negative coefficients.

Strategy Check — Variables on Both Sides 🎯

Putting It All Together: Distribute + Both Sides 🏗️

The hardest multi-step equations combine distribution with variables on both sides. Take it step by step.

Example 5: Solve $3(2x - 1) = 4x + 9$

Step 1: Distribute $6x - 3 = 4x + 9$

Step 2: Move variable terms (subtract $4x$) $2x - 3 = 9$

Step 3: Move constants (add 3) $2x = 12$

Step 4: Divide $x = 6$

Check: Left: $3(2 \cdot 6 - 1) = 3(11) = 33$. Right: $4(6) + 9 = 33$ ✓

Example 6: Solve $2(x + 4) = 3(x - 1) + 7$

Step 1: Distribute on both sides $2x + 8 = 3x - 3 + 7$

Step 2: Combine like terms on the right $2x + 8 = 3x + 4$

Step 3: Subtract $2x$ $8 = x + 4$

Step 4: Subtract 4 $4 = x$

Check: Left: $2(4+4) = 2(8) = 16$. Right: $3(4-1)+7 = 3(3)+7 = 16$ ✓

Practice: Multi-Step Equations 🧮

These are the real deal! Solve each equation for $x$.

1. $4(x - 2) = 2x + 6$

2. $3x + 5 = 7x - 11$

3. $2(3x + 1) = 5(x - 2) + 14$

Concept Check — Problem-Solving Strategy 🔍

For each equation, identify the correct FIRST step.

Part 3 Exit Quiz ✅

These require multiple steps. Take your time and work carefully.

Part 3 Complete! 🎉

You can now handle multi-step equations — the bread and butter of algebra!

Key Takeaways:

• ✅ Distribute first to eliminate parentheses
• ✅ Combine like terms on each side separately
• ✅ Move variables to one side (prefer the side that keeps the coefficient positive)
• ✅ Move constants to the other side
• ✅ Divide to isolate the variable

Next Up: Part 4 — Special Cases, Fractions, & Applications

You'll discover equations with no solution, equations with infinite solutions, and master the dreaded fraction equations!

Part 4: Special Cases, Fractions, & Real-World Applications 🌍

So far, every equation we've solved had exactly one solution. But not all equations work that way!

In this part, you'll learn:

• 📐 How to clear fractions and decimals from equations
• 🚫 Equations with no solution (contradictions)
• ♾️ Equations with infinitely many solutions (identities)
• 🌎 Translating real-world problems into equations

Clearing Fractions — The LCD Method 🔧

Fractions make equations look scary, but there's a simple trick: multiply every term by the Least Common Denominator (LCD) to eliminate all fractions at once.

Example 1: Solve $\frac{x}{3} + \frac{x}{4} = 7$

Step 1: Find the LCD of 3 and 4 → LCD = 12

Step 2: Multiply EVERY term by 12 $12 \cdot \frac{x}{3} + 12 \cdot \frac{x}{4} = 12 \cdot 7$ $4x + 3x = 84$

Step 3: Solve the simpler equation $7x = 84$ $x = 12$

Check: $\frac{12}{3} + \frac{12}{4} = 4 + 3 = 7$ ✓

Example 2: Solve $\frac{2x - 1}{5} = \frac{x + 3}{2}$

LCD of 5 and 2 → LCD = 10

Multiply every term by 10: $10 \cdot \frac{2x - 1}{5} = 10 \cdot \frac{x + 3}{2}$ $2(2x - 1) = 5(x + 3)$ $4x - 2 = 5x + 15$

Solve: $-2 - 15 = 5x - 4x$ $-17 = x$

Check: $\frac{2(-17)-1}{5} = \frac{-35}{5} = -7$. $\frac{-17+3}{2} = \frac{-14}{2} = -7$ ✓

The LCD method turns fraction equations into regular equations you already know how to solve!

Practice: Clearing Fractions 🧮

Solve each equation using the LCD method.

1. $\frac{x}{2} + \frac{x}{5} = 14$

2. $\frac{3x + 1}{4} = 7$

3. $\frac{x - 3}{2} = \frac{x + 1}{6}$

Special Cases: No Solution & Infinite Solutions 🤔

Not every equation has exactly one answer. Sometimes strange things happen when you solve…

No Solution (Contradiction):

Solve $2(x + 3) = 2x + 10$

$2x + 6 = 2x + 10$

Subtract $2x$ from both sides:

$6 = 10$

This is never true! No value of $x$ can make $6 = 10$.

Answer: No solution. The equation is a contradiction.

Visually: the lines $y = 2x + 6$ and $y = 2x + 10$ are parallel — they never intersect.

Infinite Solutions (Identity):

Solve $3(x + 2) = 3x + 6$

$3x + 6 = 3x + 6$

Subtract $3x$:

$6 = 6$

This is always true! Every value of $x$ works.

Answer: All real numbers. The equation is an identity.

Visually: $y = 3x + 6$ and $y = 3x + 6$ are the same line — they overlap everywhere.

Identify the Type of Equation 🔍

Clearing Decimals 💰

Decimal equations work the same way as fractions — multiply by a power of 10 to clear them.

Example 3: Solve $0.3x + 1.5 = 4.2$

Multiply every term by 10 (to remove one decimal place):

$3x + 15 = 42$ $3x = 27$ $x = 9$

Check: $0.3(9) + 1.5 = 2.7 + 1.5 = 4.2$ ✓

Example 4: Solve $0.05x + 0.25 = 1.75$

Multiply every term by 100 (to remove two decimal places):

$5x + 25 = 175$ $5x = 150$ $x = 30$

Check: $0.05(30) + 0.25 = 1.50 + 0.25 = 1.75$ ✓

Tip: Count the most decimal places in any term. That tells you whether to multiply by 10, 100, or 1000.

Real-World Applications 🌎

Here's where all these skills come together. The hardest part of word problems is setting up the equation. Once you have the equation, you know how to solve it!

Strategy for Word Problems:

1. Define your variable — What are you solving for?
2. Identify the relationships — What connects the quantities?
3. Write the equation — Translate English into algebra
4. Solve and check — Does your answer make sense?

Example 5: Consecutive Integers

The sum of three consecutive integers is 72. Find the integers.

Define: Let $x$ = first integer. Then: $x + 1$ = second, $x + 2$ = third.

Equation: $x + (x+1) + (x+2) = 72$

Solve: $3x + 3 = 72 \Rightarrow 3x = 69 \Rightarrow x = 23$

Answer: 23, 24, 25. Check: $23 + 24 + 25 = 72$ ✓

Example 6: Age Problem

Maria is 5 years older than twice her brother's age. If Maria is 31, how old is her brother?

Define: Let $b$ = brother's age

Equation: $2b + 5 = 31$

Solve: $2b = 26 \Rightarrow b = 13$

Answer: Her brother is 13. Check: $2(13) + 5 = 31$ ✓

Word Problem Practice 📝

Part 4 Exit Challenge 🏆

These are challenging — they combine everything from this part!

1. Solve: $\frac{2x}{3} - 4 = \frac{x}{6} + 1$ (Use the LCD method)

2. A number is tripled and then decreased by 8. The result equals the number increased by 12. Find the number.

3. Solve: $0.2x + 0.5 = 0.7x - 2$

Part 4 Complete! 🎉

You've mastered the tricky stuff — fractions, special cases, and word problems!

Key Takeaways:

• ✅ LCD method eliminates fractions — multiply every term by the LCD
• ✅ Multiply by 10, 100, etc. to clear decimals
• ✅ If you get a false statement (like $3 = 7$) → no solution
• ✅ If you get a true statement (like $5 = 5$) → infinitely many solutions
• ✅ Word problems: define the variable, set up the equation, solve, and check

Next Up: Part 5 — Mastery Challenge

Time to prove you've mastered it all! A comprehensive quiz covering everything from one-step equations to word problems. Score 80%+ to unlock Competitive Mode!

Part 5: Mastery Challenge 🏆

This is the final test! You'll face a comprehensive quiz covering everything from Parts 1–4:

• ✅ One-step equations
• ✅ Two-step equations
• ✅ Multi-step equations with distribution
• ✅ Variables on both sides
• ✅ Fraction and decimal equations
• ✅ Special cases (no solution / infinite solutions)
• ✅ Word problems

Your goal: Score 80% or higher to demonstrate mastery and unlock Competitive Mode, where you can test your skills against other students in real-time!

Take your time, show your work on paper if needed, and remember: check your answers!

Warm-Up Round 🔥

Quick review before the big quiz. Solve each equation.

1. $x - 14 = -6$

2. $-5x = 45$

3. $\frac{x}{7} + 3 = 10$

Mastery Quiz — Part A: Core Skills 📝

Answer each question carefully. These cover the foundational skills.

Mastery Quiz — Part B: Computation 🧮

These require careful multi-step work. Solve each equation.

1. $5(x + 3) = 2(x + 6) + 12$

2. $\frac{3x + 2}{4} = \frac{x - 6}{2}$

3. $0.4x - 1.2 = 0.1x + 0.6$

Mastery Quiz — Part C: Advanced Concepts 🧠

These test deeper understanding — not just mechanics.

Final Round — Prove Your Mastery 💪

These are the toughest problems. Solve each equation.

1. $2(3x + 1) - (x - 3) = 3(x + 5)$

2. $\frac{5x - 3}{6} + 1 = \frac{x + 3}{2}$

3. $0.25(x - 4) + 0.5x = 2.75$

🎉 Congratulations — You've Mastered Solving Linear Equations! 🎉

You've completed all 5 parts and proven your mastery of one of algebra's most important skills.

What You've Learned:

• ✅ One-step equations (addition, subtraction, multiplication, division)
• ✅ Two-step equations (reverse order of operations)
• ✅ Multi-step equations (distribute, combine like terms)
• ✅ Variables on both sides
• ✅ Clearing fractions and decimals
• ✅ Special cases (no solution & infinite solutions)
• ✅ Setting up and solving word problems

🏆 Competitive Mode Unlocked!

You're now ready to compete! Head to Competitive Mode to test your equation-solving speed against other students in real-time challenges.

Keep Practicing: The more equations you solve, the faster and more accurate you'll become. These skills are the foundation for everything that comes next in algebra!