Solving One-Step Equations

⚖️ Solving One-Step Equations

Part 1 of 5 — The Balance Idea & Inverse Operations

Topics in This Part

🔑 Key Concept: An equation is a balance. Whatever you do to one side, you must do to the other side to keep it balanced. Solving means getting the variable alone on one side.

What Is an Equation?

An equation is a math sentence that says two things are equal. It always has an equals sign ($=$).

$x + 4 = 9$

The letter $x$ is the variable — it stands for a number we don't know yet. Solving the equation means finding the value of the variable that makes the sentence true.

💡 One-step equations need just one move to solve. That single move always undoes whatever is happening to the variable.

Concept Check 🎯

The Balance Scale

Picture an equation as a balance scale. The $=$ sign is the middle, and both sides weigh the same.

$\underbrace{x + 4}_{\text{left pan}} \;=\; \underbrace{9}_{\text{right pan}}$

⚖️ The Golden Rule: Whatever you do to one side, do the exact same thing to the other side. This keeps the scale balanced.

If you take 4 away from the left pan, you must take 4 away from the right pan too:

$x + 4 - 4 = 9 - 4 \;\Rightarrow\; x = 5$

The scale stays level, and now $x$ sits alone. We solved it.

Try One 🧮

Use the balance idea on $x + 4 = 9$. Take $4$ away from both sides.

1) $9 - 4 = \,?$ (the right pan after removing 4) 2) So $x = \,?$

Inverse Operations

To get the variable alone, you undo the operation attached to it. The undo move is called an inverse operation — every operation has an opposite.

Examples:

• $x + 4 = 9$ → the variable is being added to, so subtract $4$.
• $6y = 24$ → the variable is being multiplied, so divide by $6$.

🔑 The plan for every one-step equation: find the operation on the variable, then do its inverse to both sides.

Pick the Inverse 🔽

For each equation, choose the operation you'd use to get the variable alone.

⚖️ Solving One-Step Equations

Part 2 of 5 — Addition & Subtraction Equations

🔑 The Idea: When a number is added to the variable, subtract it from both sides. When a number is subtracted, add it to both sides.

Equations with Addition

When the variable has a number added to it, subtract that number from both sides.

Worked Example: $x + 7 = 12$

The variable has $+7$ attached, so subtract $7$ from both sides:

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

✅ Check: Put $5$ back in: $5 + 7 = 12$ ✓ True! So $x = 5$ is correct.

Equations with Subtraction

When a number is subtracted from the variable, add that number to both sides.

Worked Example: $n - 4 = 10$

The variable has $-4$ attached, so add $4$ to both sides:

$n - 4 + 4 = 10 + 4$ $n = 14$

✅ Check: $14 - 4 = 10$ ✓

Worked Example: $y - 9 = 3$

$y - 9 + 9 = 3 + 9 \;\Rightarrow\; y = 12$

💡 Notice the pattern: subtraction in the equation is undone with addition. Always do the opposite.

Concept Check 🎯

Solve It 🧮

Solve each equation for the variable. Enter just the number.

1) $x + 9 = 16,\quad x = \,?$ 2) $n - 5 = 11,\quad n = \,?$ 3) $y + 13 = 13,\quad y = \,?$

Choose the First Move 🔽

Pick the correct first step and the solution for each equation.

⚖️ Solving One-Step Equations

Part 3 of 5 — Multiplication & Division Equations

🔑 The Idea: When the variable is multiplied by a number, divide both sides by it. When the variable is divided, multiply both sides by it.

Equations with Multiplication

$6y = 24$ means "$6$ times $y$." A number written right next to a variable means multiply.

To undo multiplication, divide both sides by that number.

Worked Example: $6y = 24$

$\frac{6y}{6} = \frac{24}{6}$ $y = 4$

✅ Check: $6 \times 4 = 24$ ✓

Worked Example: $5x = 35$

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

⚠️ Watch out: $6y$ does not mean "$6$ plus $y$." It means "$6$ times $y$," so you divide to undo it — never subtract.

Equations with Division

$\dfrac{x}{3} = 5$ means "$x$ divided by $3$." To undo division, multiply both sides by that number.

Worked Example: $\dfrac{x}{3} = 5$

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

✅ Check: $\dfrac{15}{3} = 5$ ✓

Worked Example: $\dfrac{m}{4} = 8$

$\frac{m}{4} \times 4 = 8 \times 4 \;\Rightarrow\; m = 32$

💡 Multiplication and division are opposites, just like addition and subtraction. Spot the operation, then do its inverse.

Concept Check 🎯

Match the Move 🔽

Choose how to undo each equation, then the solution.

Solve It 🧮

Solve each equation. Enter just the number.

1) $7x = 42,\quad x = \,?$ 2) $\dfrac{n}{5} = 6,\quad n = \,?$ 3) $3y = 27,\quad y = \,?$

⚖️ Solving One-Step Equations

Part 4 of 5 — Checking Answers & Word Problems

🔑 Big Skill: Every solution can be checked by substituting it back in. And many real-life problems become one-step equations once you translate the words into math.

Always Check Your Answer

To check, substitute your answer back into the original equation. If both sides are equal, you're right.

Example: Solve and check $x + 12 = 20$

$x = 20 - 12 = 8$

Check: replace $x$ with $8$: $8 + 12 = 20 \;\checkmark$

Both sides equal $20$, so $x = 8$ is correct.

⚠️ Checking catches mistakes! If your answer makes the two sides unequal, you used the wrong inverse — go back and try again.

Concept Check 🎯

Turning Words into Equations

Word problems hide a one-step equation. Watch for these phrases:

Example

"Maya had some stickers. After she got $6$ more, she had $15$. How many did she start with?"

Let $s$ = the starting number. "Got $6$ more" means $+6$, "had $15$" means $= 15$:

$s + 6 = 15 \;\Rightarrow\; s = 15 - 6 = 9$

Maya started with $9$ stickers.

Word Problems 🧮

Write a one-step equation in your head, then enter the answer (just the number).

1) A number times $4$ equals $32$. What is the number? 2) After spending $8, Liam had $5 left. How much did he have at first? (dollars) 3) A rope is split into $3$ equal pieces, each $7$ ft long. How long was the rope? (ft)

Set Up the Equation 🔽

Match each story to the correct one-step equation (let $n$ be the unknown).

⚖️ Solving One-Step Equations

Part 5 of 5 — Mixed Practice & Mastery Check

You can now solve one-step equations with all four operations, check your answers, and translate word problems. Let's put it all together.

Quick Reference

⚖️ Remember the Golden Rule: do the same inverse operation to both sides, then check by substituting your answer back in.

Mixed Practice 🎯

Final Drill 🧮

One of each kind. Enter just the number.

1) $x + 11 = 30,\quad x = \,?$ 2) $9x = 72,\quad x = \,?$ 3) $\dfrac{x}{8} = 5,\quad x = \,?$ 4) $x - 6 = 6,\quad x = \,?$

Exit Quiz ✅

Answer all three to finish the lesson.