Solving One-Step Equations

⚖️ Solving One-Step Equations

Part 1 of 5 — Equations, Balance & Inverse Operations

Topics in This Part

🔑 Key Concept: An equation is a statement that two things are equal. To solve one, you find the value of the variable that makes the statement true — and you do it by keeping both sides perfectly balanced.

What Is an Equation?

An equation has an equals sign $=$ with an expression on each side:

$x + 4 = 9$

The letter $x$ is the variable — an unknown number we want to find. Solving the equation means finding the value of $x$ that makes the left side truly equal the right side.

For $x + 4 = 9$, the answer is $x = 5$, because $5 + 4 = 9$. ✓

💡 A one-step equation needs exactly one operation to undo. $x + 4 = 9$ has just an addition to undo, so it takes one step. We'll build all four types: $+$, $-$, $\times$, and $\div$.

Equation vs. Expression

Concept Check 🎯

The Balance Idea

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

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

🔑 The Golden Rule of Equations: Whatever you do to one side, you must do to the other side. This keeps the scale balanced and the equation true.

If you remove $4$ from the left pan only, the scale tips — the sides are no longer equal. But if you remove $4$ from both pans, it stays level:

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

That single balanced move is the whole secret to solving equations.

Keep It Balanced 🔽

To keep each equation balanced, choose what you must do to the other side.

Inverse Operations

To get the variable alone, you undo the operation attached to it. The tool for undoing is an inverse operation — an operation that reverses another.

The plan for every one-step equation:

1. Look at what is being done to the variable.
2. Do the inverse to both sides.
3. The variable ends up alone — that's your answer.

💡 Example: In $x + 4 = 9$, the $4$ is added, so you subtract $4$ from both sides. In $5x = 20$, the $x$ is multiplied by $5$, so you divide both sides by $5$.

You now have the two ideas the whole topic rests on: balance and inverse operations. In Part 2 we put them to work with addition and subtraction.

Pick the Inverse 🎯

⚖️ 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. Inverse operation, both sides, done.

Equations with Addition

When something is added to the variable, undo it by subtracting from both sides.

Worked Example: $x + 6 = 15$

The $6$ is added to $x$, so subtract $6$ from both sides:

$x + 6 - 6 = 15 - 6$ $x = 9$

✅ Check: Put $9$ back in: $9 + 6 = 15$ ✓

Worked Example: $13 = n + 8$

The variable can be on the right — that's fine. Subtract $8$ from both sides:

$13 - 8 = n + 8 - 8$ $5 = n$

💡 $5 = n$ means exactly the same as $n = 5$. The equals sign reads both directions.

Concept Check 🎯

Equations with Subtraction

When a number is subtracted from the variable, undo it by adding to both sides.

Worked Example: $x - 5 = 12$

The $5$ is subtracted, so add $5$ to both sides:

$x - 5 + 5 = 12 + 5$ $x = 17$

✅ Check: $17 - 5 = 12$ ✓

Worked Example: $y - 9 = -3$

A negative result is allowed. Add $9$ to both sides:

$y - 9 + 9 = -3 + 9$ $y = 6$

⚠️ Watch the sign of the answer side. $-3 + 9 = 6$, not $-12$. Adding a positive moves you up the number line: from $-3$, count up $9$ to land on $6$.

Solve It 🧮

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

1) $x + 4 = 11 \;\Rightarrow\; x = \,?$ 2) $x - 8 = 5 \;\Rightarrow\; x = \,?$ 3) $x + 10 = 3 \;\Rightarrow\; x = \,?$ (answer may be negative)

One Rule, Two Directions

Notice that addition and subtraction equations use the same logic — just pointed the opposite way:

• See a $+$? Subtract it from both sides.
• See a $-$? Add it to both sides.

💡 A quick way to think about it: "move" the number to the other side and flip its sign. In $x + 6 = 15$, the $+6$ becomes $-6$ on the right: $x = 15 - 6 = 9$. That shortcut is really just the inverse-operation rule in disguise.

Let's walk through the reasoning one step at a time.

Order the Solution 🔽

You are solving $x - 6 = 10$. Choose what happens at each stage.

⚖️ 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

$5x$ means "$5$ times $x$." The $5$ is a coefficient stuck to the variable by multiplication. Undo it by dividing both sides by that coefficient.

Worked Example: $5x = 20$

$\frac{5x}{5} = \frac{20}{5}$ $x = 4$

✅ Check: $5 \cdot 4 = 20$ ✓

Worked Example: $-3x = 21$

Divide both sides by $-3$. A negative divided into a positive gives a negative:

$\frac{-3x}{-3} = \frac{21}{-3}$ $x = -7$

✅ Check: $-3 \cdot (-7) = 21$ ✓

⚠️ Sign rule: dividing by a negative coefficient flips the sign of the answer. Track that minus sign carefully.

Concept Check 🎯

Equations with Division

$\dfrac{x}{4}$ means "$x$ divided by $4$." Undo the division by multiplying both sides by that number.

Worked Example: $\dfrac{x}{4} = 6$

$\frac{x}{4} \cdot 4 = 6 \cdot 4$ $x = 24$

✅ Check: $\dfrac{24}{4} = 6$ ✓

Worked Example: $\dfrac{n}{3} = -5$

Multiply both sides by $3$:

$n = -5 \cdot 3 = -15$

💡 Key contrast: if the variable is divided by a number, you multiply to undo it. If it is multiplied, you divide. Always do the opposite.

Solve It 🧮

Solve each equation for the variable.

1) $7x = 42 \;\Rightarrow\; x = \,?$ 2) $\dfrac{x}{5} = 9 \;\Rightarrow\; x = \,?$ 3) $-6x = 30 \;\Rightarrow\; x = \,?$ (answer may be negative)

Spotting Which Inverse to Use

Before solving, ask one question: how is the variable attached to its number?

⚠️ Don't mix up the four operations. A coefficient like the $7$ in $7x$ is not added — it's multiplied. Adding or subtracting it would be the wrong inverse. Read the equation carefully first, then pick the matching inverse.

Match the Move 🔽

Choose the single operation that solves each equation.

⚖️ Solving One-Step Equations

Part 4 of 5 — Fractions, Decimals & Checking Your Work

🔑 The Idea: The exact same rules — inverse operation, both sides — work when the numbers are fractions or decimals. And checking your answer catches mistakes every time.

Equations with Decimals

Decimals follow the same rules. The operation tells you what to undo.

Worked Example: $x + 2.5 = 6$

The $2.5$ is added, so subtract $2.5$ from both sides:

$x = 6 - 2.5 = 3.5$

✅ Check: $3.5 + 2.5 = 6$ ✓

Worked Example: $0.4x = 2$

The $x$ is multiplied by $0.4$, so divide both sides by $0.4$:

$x = \frac{2}{0.4} = 5$

💡 Dividing by a decimal less than $1$ makes the answer bigger — here $2 \div 0.4 = 5$. Check: $0.4 \cdot 5 = 2$ ✓

Decimal Drill 🧮

Solve each equation. Decimals are fine in your answer.

1) $x - 1.2 = 4 \;\Rightarrow\; x = \,?$ 2) $x + 3.5 = 5 \;\Rightarrow\; x = \,?$ 3) $2x = 7 \;\Rightarrow\; x = \,?$ (decimal is fine)

Equations with Fractions

Adding or Subtracting a Fraction

Worked Example: $x - \dfrac{1}{4} = \dfrac{3}{4}$

Add $\dfrac{1}{4}$ to both sides. With a common denominator, just add the numerators:

$x = \frac{3}{4} + \frac{1}{4} = \frac{4}{4} = 1$

✅ Check: $1 - \dfrac{1}{4} = \dfrac{3}{4}$ ✓

A Fraction Times the Variable

Worked Example: $\dfrac{2}{3}x = 8$

To undo multiplication by a fraction, multiply by its reciprocal (flip it). The reciprocal of $\dfrac{2}{3}$ is $\dfrac{3}{2}$:

$\frac{3}{2} \cdot \frac{2}{3}x = \frac{3}{2} \cdot 8$ $x = \frac{3 \cdot 8}{2} = \frac{24}{2} = 12$

✅ Check: $\dfrac{2}{3} \cdot 12 = \dfrac{24}{3} = 8$ ✓

💡 Reciprocal trick: $\dfrac{a}{b}x = c$ is solved by $x = \dfrac{b}{a}\cdot c$. Flipping the fraction turns "divide" trouble into easy multiplication.

Concept Check 🎯

Always Check Your Answer

Checking means substituting your answer back into the original equation and confirming both sides match. It catches sign slips and arithmetic errors instantly.

Example: did we solve $x + 9 = 4$ correctly with $x = -5$?

Substitute $x = -5$ into the original:

$-5 + 9 = 4 \;?\quad\Rightarrow\quad 4 = 4 \;✓$

It checks, so $x = -5$ is correct.

⚠️ Common trap: if your check gives a false statement like $7 = 4$, you made an error — usually a dropped negative sign or an arithmetic slip. Re-solve and check again.

You can now solve any one-step equation with whole numbers, negatives, decimals, or fractions — and verify it. In Part 5 we turn real-world sentences into equations and finish with an Exit Quiz.

Check the Work 🔽

For each equation, a proposed solution is given. Decide whether it checks (substitute it back).

⚖️ Solving One-Step Equations

Part 5 of 5 — Word Problems, Mixed Practice & Mastery Check

You can now solve one-step equations of every kind. Let's translate real sentences into equations, mix all four operations together, and finish with an Exit Quiz.

Turning Words into Equations

Real problems arrive as sentences. Translate the words into symbols, then solve in one step.

Worked Example

"A number increased by $7$ is $20$. Find the number."

Translate: $x + 7 = 20$. Subtract $7$ from both sides:

$x = 20 - 7 = 13$

✅ Check: $13 + 7 = 20$ ✓

Worked Example

"Four times a number equals $36$. Find the number."

Translate: $4x = 36$. Divide both sides by $4$:

$x = 36 \div 4 = 9 \quad ✓$

Translate & Solve 🎯

Three Steps for Any Word Problem

1. Name the unknown. Let $x$ stand for "the number" you're looking for.
2. Translate the sentence phrase-by-phrase into symbols, with "is" or "equals" becoming $=$.
3. Solve the one-step equation and check by re-reading the sentence with your answer in place.

💡 Watch the key words: increased by and sum mean $+$; decreased by and less than mean $-$; times, product, and twice mean $\times$; split, shared equally, and quotient mean $\div$.

Now try a few on your own.

Word-Problem Drill 🧮

Translate each sentence to a one-step equation and solve. Enter just the number.

1) A number increased by $12$ is $30$. The number $= \,?$ 2) A number split into $5$ equal parts gives $4$. The number $= \,?$ 3) Twice a number is $-18$. The number $= \,?$ (answer may be negative)

Quick Reference

⚠️ Always: keep both sides balanced, watch negative signs, and check by substituting your answer into the original equation.

Mixed Practice 🎯

Exit Quiz ✅

Answer all three to finish the lesson.