Linear Equations

⚖️ Linear Equations

Part 1 of 5 — Concept Introduction

A linear equation is an equation whose variable only ever has an exponent of $1$ — no $x^2$, no $x^3$, and no $x$ hiding in a denominator. When you graph a linear equation in two variables, you always get a perfectly straight line, which is exactly where the name linear comes from.

In Grade 8 we focus on solving linear equations in one variable. Solving means finding the single value of $x$ that makes the two sides equal. Think of the equals sign as the middle of a balance scale: whatever you do to one side, you must do to the other to keep it balanced.

Linear vs. Not Linear

If you can write it without exponents above $1$ and without a variable in the denominator, it is linear.

The Golden Rule of Balance

To solve $x + 4 = 9$, you want $x$ all by itself. Right now $4$ is being added to $x$, so you undo it with the opposite operation — subtraction:

$x + 4 - 4 = 9 - 4$ $x = 5$

We subtracted $4$ from both sides so the scale stays balanced. These pairs of opposite operations are called inverse operations:

• Addition ↔ Subtraction
• Multiplication ↔ Division

Example: Solve $3x = 12$. Here $x$ is multiplied by $3$, so undo it by dividing both sides by $3$:

$\frac{3x}{3} = \frac{12}{3} \quad\Rightarrow\quad x = 4$

💡 Key idea: To get a variable alone, undo whatever is happening to it using the inverse operation — and always do it to both sides.

Two-Step Equations

Most equations need two moves. The trick is to undo operations in reverse order — strip away addition/subtraction first, then multiplication/division.

Example: Solve $2x + 3 = 11$.

1. Undo the $+3$ first → subtract $3$ from both sides: $2x = 8$
2. Undo the $\times 2$ → divide both sides by $2$: $x = 4$

Check: Put $x = 4$ back in: $2(4) + 3 = 8 + 3 = 11$. ✅ It works!

Always finish by checking: substitute your answer into the original equation and confirm both sides are equal.

Concept Check 🎯

Make sure you can spot which operation to use.

📝 Worked Examples

Part 2 of 5 — Variables on Both Sides

When the variable appears on both sides of the equals sign, your first job is to collect all the variable terms on one side and all the plain numbers on the other. You do this with the same balance rule: subtract a term from both sides to make it disappear from one side.

Example 1: Solve $5x = 2x + 9$

1. Get the $x$'s together. Subtract $2x$ from both sides: $5x - 2x = 2x + 9 - 2x \quad\Rightarrow\quad 3x = 9$
2. Divide both sides by $3$: $x = 3$
3. Check: $5(3) = 15$ and $2(3) + 9 = 15$. ✅ Both sides equal $15$.

Example 2: Solve $4x + 1 = 2x + 7$

1. Subtract $2x$ from both sides to gather the variables on the left: $2x + 1 = 7$
2. Subtract $1$ from both sides to gather the numbers on the right: $2x = 6$
3. Divide both sides by $2$: $x = 3$
4. Check: $4(3) + 1 = 13$ and $2(3) + 7 = 13$. ✅

Strategy reminder: move the smaller variable term to keep your coefficient positive — it makes the arithmetic easier.

Your Turn 🧮

Solve the equation $6x + 2 = 4x + 10$ one step at a time.

1. After subtracting $4x$ from both sides, what number is the new coefficient of $x$? (For example, if you get $2x$, type 2.)

2. After also subtracting $2$ from both sides, what number is left on the right side? (You should have (your coefficient)$x =$ that number.)

3. Finally, what is the value of $x$?

🧭 Guided Practice

Part 3 of 5 — Guided Practice

Work each equation carefully. Remember: collect the variables on one side, the numbers on the other, then divide.

Choose the Right Move 🔍

You are solving $9x + 5 = 4x + 25$. Pick the correct step at each stage.

🌍 Application & Word Problems

Part 4 of 5 — Linear Equations in Real Life

Equations with variables on both sides show up whenever you compare two plans and ask, "When are they equal?"

The Two Gyms Problem 🏋️

Gym A charges a $20 sign-up fee plus $10 per month. Gym B charges no sign-up fee but $15 per month. After how many months do they cost the same?

Let $x$ = number of months. Write the cost of each gym:

• Gym A: $10x + 20$
• Gym B: $15x$

Set them equal because we want the costs to match:

$10x + 20 = 15x$

Now solve. Subtract $10x$ from both sides: $20 = 5x$. Divide by $5$: $x = 4$.

Answer: After 4 months both gyms cost the same — $60 each. The variable-on-both-sides equation came straight from a real decision!

Word Problem Practice 🧮

Two taxi companies are competing for your ride:

• Yellow Cab: $5 base fare plus $2 per mile → cost $= 2x + 5$
• Blue Cab: $3 per mile, no base fare → cost $= 3x$

You want to know after how many miles the two cabs cost the same, so set $2x + 5 = 3x$.

1. After subtracting $2x$ from both sides, what number equals $x$ (the number of miles where the costs match)?

2. What is the total cost of either cab at that number of miles? (In dollars, type a single number.)

3. For a $10$-mile trip, which is cheaper — type 2 for Yellow Cab's cost $2(10)+5$ or 3 for Blue Cab's cost $3(10)$, whichever is the smaller total. (Type the smaller of the two totals as a single number.)

Word Problem Check 📋

🏆 Review & Challenge

Part 5 of 5 — Putting It All Together

You can now solve any one-variable linear equation — even when the variable is on both sides. Here is the complete game plan:

💡 Special cases: If the variables cancel and you get a true statement like $5 = 5$, there are infinitely many solutions. If you get a false statement like $5 = 8$, there is no solution.

Challenge Round 🎯

Mix everything you have learned. Take your time and check each answer.