Solving Linear Equations

⚖️ Solving Linear Equations

Part 1 of 5 — The Balance Idea & One-Step Equations

Topics in This Part

🔑 Key Concept: An equation is a balanced scale. Whatever you do to one side, you must do to the other side too — that single rule powers everything in this course.

What Does "Solving" Mean?

A linear equation is a statement that two expressions are equal, like

$x + 5 = 12$

To solve it means to find the value of the variable that makes the statement true. Here the answer is $x = 7$, because $7 + 5 = 12$. ✓

The variable is usually alone on one side once you're done:

$x = 7$

💡 Goal of every problem: get the variable by itself on one side of the equals sign. We call this isolating the variable.

Checking a Solution

You can always test an answer by substituting it back in. Is $x = 4$ a solution of $2x = 8$?

$2(4) = 8 \quad\Rightarrow\quad 8 = 8 \;\checkmark$

Yes! Checking is a free way to catch mistakes — get in the habit.

Concept Check 🎯

Keep the Equation Balanced

Think of $=$ as the center of a balance scale. The two sides weigh the same. If you add or remove weight from one side, you must do the same to the other, or the scale tips.

⚠️ The #1 beginner mistake is changing only one side. If you subtract $5$ from the left, you must subtract $5$ from the right.

Inverse Operations

To isolate the variable, you "undo" what's attached to it using the opposite operation:

One-Step Equations: Worked Examples

Example A — undo addition

$x + 8 = 15$

The $+8$ is in the way. Undo it by subtracting $8$ from both sides:

$x + 8 - 8 = 15 - 8 \quad\Rightarrow\quad x = 7$

✅ Check: $7 + 8 = 15$ ✓

Example B — undo subtraction

$x - 6 = -2$

Undo the $-6$ by adding $6$ to both sides:

$x - 6 + 6 = -2 + 6 \quad\Rightarrow\quad x = 4$

Example C — undo multiplication

$5x = 35$

The $x$ is multiplied by $5$. Undo it by dividing both sides by $5$:

$\frac{5x}{5} = \frac{35}{5} \quad\Rightarrow\quad x = 7$

Example D — undo division

$\frac{x}{4} = 3$

Undo dividing by $4$ by multiplying both sides by $4$:

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

Pick the Move 🔽

For each equation, choose the single operation that isolates $x$.

Before You Practice

Every one-step equation follows the same plan:

1. Spot what's done to the variable.
2. Apply the inverse operation to both sides.
3. Check by substituting your answer back in.

🔑 One operation, both sides, then verify. Master this and two-step equations (Part 2) are just two of these in a row.

Solve One-Step Equations 🧮

Solve for $x$. Enter just the number.

1) $x + 12 = 20 \;\Rightarrow\; x = \,?$ 2) $x - 7 = -3 \;\Rightarrow\; x = \,?$ 3) $6x = 54 \;\Rightarrow\; x = \,?$ 4) $\dfrac{x}{5} = 8 \;\Rightarrow\; x = \,?$

⚖️ Solving Linear Equations

Part 2 of 5 — Two-Step Equations

🔑 The Idea: When a variable has two things done to it (like "times $3$, then plus $4$"), undo them in the reverse order — addition/subtraction first, then multiplication/division.

Undo in Reverse Order

An equation like $3x + 4 = 19$ was built in this order:

$x \;\xrightarrow{\;\times 3\;}\; 3x \;\xrightarrow{\;+\,4\;}\; 3x + 4$

To get back to $x$, reverse the steps — undo the last thing first:

1. Undo $+4$ → subtract $4$
2. Undo $\times 3$ → divide by $3$

💡 Memory trick: it's like getting dressed and undressed. You put socks on before shoes, so you take shoes off before socks. Reverse order!

Worked Examples

Example A — $3x + 4 = 19$

Step 1 — subtract $4$ from both sides: $3x + 4 - 4 = 19 - 4 \quad\Rightarrow\quad 3x = 15$

Step 2 — divide both sides by $3$: $\frac{3x}{3} = \frac{15}{3} \quad\Rightarrow\quad x = 5$

✅ Check: $3(5) + 4 = 15 + 4 = 19$ ✓

Example B — $\dfrac{x}{2} - 6 = 1$

Step 1 — add $6$ to both sides: $\frac{x}{2} = 7$

Step 2 — multiply both sides by $2$: $x = 14$

Example C — a negative coefficient: $-2x + 9 = 1$

Step 1 — subtract $9$: $\;-2x = -8$ Step 2 — divide by $-2$: $\;x = 4$

⚠️ When you divide by a negative, the sign flips. Here $\dfrac{-8}{-2} = 4$ (positive).

Concept Check 🎯

Show Every Step in Writing

Two-step problems are easy to slip on if you do them in your head. Write each line under the last so the chain of reasoning is visible:

$4x - 5 = 11$ $4x = 16 \quad (\text{add } 5)$ $x = 4 \quad (\text{divide by } 4)$

💡 Lining up your work this way makes mistakes jump out — and earns full credit on tests. Let's trace one stage by stage.

Trace the Steps 🔽

You're solving $4x - 5 = 11$. Fill in each stage.

A Note on Negative Answers

Two-step equations often produce negative solutions — that's perfectly normal. Just apply the rules carefully with signs:

$-3x + 1 = 10 \;\Rightarrow\; -3x = 9 \;\Rightarrow\; x = \frac{9}{-3} = -3$

And answers don't have to be whole numbers. If dividing doesn't come out evenly, leave a fraction or use a decimal:

$6x = 9 \;\Rightarrow\; x = \frac{9}{6} = \frac{3}{2} = 1.5$

Now try a mixed set on your own.

Solve Two-Step Equations 🧮

Solve for $x$. Fractions like $-5/2$ or decimals are fine.

1) $2x + 5 = 17 \;\Rightarrow\; x = \,?$ 2) $\dfrac{x}{3} + 4 = 10 \;\Rightarrow\; x = \,?$ 3) $-3x + 1 = 10 \;\Rightarrow\; x = \,?$ 4) $6x - 7 = 2 \;\Rightarrow\; x = \,?$ (fraction or decimal)

⚖️ Solving Linear Equations

Part 3 of 5 — Distributing & Combining Like Terms

🔑 The Idea: Before you isolate the variable, clean up each side: distribute through parentheses and combine like terms so every side is as simple as possible.

The Distributive Property

To multiply a number across a sum, multiply it by each term inside the parentheses:

$a(b + c) = ab + ac$

⚠️ Watch the signs! $-4(x+2) = -4x - 8$, and the minus in $-(x-6)$ flips both terms: $-x + 6$.

Combining Like Terms

Like terms have the same variable part. Add their coefficients:

$5x + 2x = 7x \qquad 8 - 3 = 5 \qquad 5x + 8 - 3x - 3 = 2x + 5$

You can only combine $x$-terms with $x$-terms, and numbers with numbers.

Concept Check 🎯

Variables on Both Sides

When $x$ appears on both sides, move all the variables to one side (and all the numbers to the other) using inverse operations.

Worked Example — $5x + 3 = 2x + 18$

Step 1 — subtract $2x$ from both sides (gather $x$ on the left): $5x - 2x + 3 = 18 \quad\Rightarrow\quad 3x + 3 = 18$

Step 2 — subtract $3$: $3x = 15$

Step 3 — divide by $3$: $x = 5$

✅ Check: Left $= 5(5)+3 = 28$. Right $= 2(5)+18 = 28$. ✓

💡 Tip: Move the smaller variable term to avoid negatives. Here $2x < 5x$, so we subtracted $2x$.

Putting It Together — Distribute First, Then Solve

Worked Example — $3(x + 2) = 21$

Step 1 — distribute the $3$: $3x + 6 = 21$

Step 2 — subtract $6$: $\;3x = 15$ Step 3 — divide by $3$: $\;x = 5$

Worked Example — $2(x - 4) = x + 1$

Step 1 — distribute: $\;2x - 8 = x + 1$ Step 2 — subtract $x$: $\;x - 8 = 1$ Step 3 — add $8$: $\;x = 9$

✅ Check: $2(9-4) = 2(5) = 10$, and $9 + 1 = 10$. ✓

Solve $6x - 1 = 4x + 9$ 🔽

Choose what happens at each stage.

The General Multi-Step Plan

For any linear equation, work through these in order:

1. Distribute to clear parentheses.
2. Combine like terms on each side.
3. Gather variables on one side, numbers on the other.
4. Isolate the variable (divide by its coefficient).
5. Check your answer.

💡 Not every problem needs every step — but doing them in this order never fails. Try the plan below.

Multi-Step Practice 🧮

Solve for $x$. Enter just the number.

1) $4(x + 3) = 24 \;\Rightarrow\; x = \,?$ 2) $7x + 2 = 3x + 18 \;\Rightarrow\; x = \,?$ 3) $5(x - 1) = 2x + 7 \;\Rightarrow\; x = \,?$

⚖️ Solving Linear Equations

Part 4 of 5 — Fractions, Decimals & Special Cases

🔑 The Idea: Fractions and decimals don't change the rules — you can clear them with one smart multiplication. And some equations have no solution or infinitely many solutions; you'll learn to spot both.

Clearing Fractions

To erase fractions, multiply every term on both sides by the least common denominator (LCD).

Worked Example — $\dfrac{x}{2} + \dfrac{1}{3} = \dfrac{5}{6}$

The denominators are $2, 3, 6$. Their LCD is $6$. Multiply every term by $6$:

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

$3x + 2 = 5$

Now it's an ordinary two-step equation:

$3x = 3 \quad\Rightarrow\quad x = 1$

✅ Check: $\dfrac{1}{2} + \dfrac{1}{3} = \dfrac{3}{6} + \dfrac{2}{6} = \dfrac{5}{6}$ ✓

💡 Multiplying by the LCD turns a messy fraction equation into a clean whole-number one in a single step.

Clearing Decimals

Same trick: multiply every term by a power of $10$ ($10$, $100$, …) big enough to make all decimals whole.

Worked Example — $0.2x + 1.5 = 2.1$

Each number has one decimal place, so multiply every term by $10$:

$2x + 15 = 21$

$2x = 6 \quad\Rightarrow\quad x = 3$

✅ Check: $0.2(3) + 1.5 = 0.6 + 1.5 = 2.1$ ✓

You could also solve with the decimals left in — clearing them is just optional cleanup.

Concept Check 🎯

Special Cases: No Solution & Infinitely Many

Most equations have exactly one solution. But after simplifying, sometimes the variable disappears. What's left tells you the answer.

No Solution — a false statement

$2x + 5 = 2x + 9$

Subtract $2x$ from both sides:

$5 = 9 \quad (\text{false!})$

Since $5 = 9$ is never true, there is no solution. No value of $x$ works.

Infinitely Many Solutions — a true statement

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

Distribute the left side: $3x + 6 = 3x + 6$. Subtract $3x$:

$6 = 6 \quad (\text{always true!})$

Since $6 = 6$ is always true, every number is a solution — infinitely many solutions.

Classify the Equation 🔽

Simplify each equation in your head, then choose how many solutions it has.

Putting Parts 1–4 Together

You now have a full toolkit:

After cleanup, isolate the variable as usual — or, if it cancels, classify the equation as no solution or infinitely many. Time to combine it all.

Fractions, Decimals & Classifying 🧮

For #1–2 enter the numeric solution. For #3 enter the number of solutions as a digit: 1 for one, 0 for none.

1) $\dfrac{x}{3} + 2 = 5 \;\Rightarrow\; x = \,?$ 2) $0.4x - 1.2 = 0.8 \;\Rightarrow\; x = \,?$ 3) How many solutions does $7x + 4 = 7x - 2$ have? (enter $0$ or $1$)

⚖️ Solving Linear Equations

Part 5 of 5 — Word Problems & Mastery Check

You can now solve one-step, two-step, and multi-step equations, clear fractions and decimals, and recognize special cases. The last skill is the most useful in real life: turning a sentence into an equation.

Translating Words into Equations

Pick a letter for the unknown, then translate phrase by phrase.

Worked Example

"Four more than three times a number is $25$. Find the number."

Translate: $\;3x + 4 = 25$.

Solve: $\;3x = 21 \Rightarrow x = 7$.

✅ Check: $3(7) + 4 = 21 + 4 = 25$ ✓ — the number is $7$.

Translate & Solve 🎯

A Geometry Application

Equations show up everywhere — here's a perimeter problem.

"A rectangle's length is $3$ cm more than its width. Its perimeter is $26$ cm. Find the width."

Let the width be $w$. Then the length is $w + 3$. Perimeter is $2(\text{length}) + 2(\text{width})$:

$2(w + 3) + 2w = 26$

Distribute and combine:

$2w + 6 + 2w = 26 \quad\Rightarrow\quad 4w + 6 = 26$

$4w = 20 \quad\Rightarrow\quad w = 5$

✅ Check: width $= 5$, length $= 8$, perimeter $= 2(8) + 2(5) = 16 + 10 = 26$ ✓

💡 Strategy: (1) name the unknown, (2) write expressions for the other quantities, (3) build the equation, (4) solve, (5) check it makes sense.

Word-Problem Practice 🧮

Set up an equation and solve. Enter just the number.

1) Five more than twice a number is $23$. What is the number? 2) Maya has $40. She buys a backpack for $16 and then $x$ notebooks at $3 each, spending all $40. How many notebooks? 3) Three consecutive integers sum to $48$. What is the smallest one? (Hint: $x + (x+1) + (x+2) = 48$.)

Quick Reference

⚠️ Always do the same thing to both sides, and check your answer by substituting it back in.

Exit Quiz ✅

Answer all three to finish the lesson.