⚡ Topic Learning Kit

Simple Equations

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💡 Key Idea

An Equation Is a Balanced Scale

Both sides must always stay equal.

A simple equation says two expressions are equal. Whatever you do to one side, you must do to the other — just like keeping a balance scale level.

↳ Treat an equation like a balance: every operation must be mirrored on both sides.

📖 Definition

Key Terms You Must Know

Speak the language of equations.

↳ Every equation has a variable, two sides, and a solution that balances them.

➗ Formula

The General Form of a Simple Equation

One pattern covers them all.

Solving means finding $x$ such that the equation holds. For example, $2x + 3 = 11$ gives $x = 4$.

↳ Most simple equations reduce to the form $ax + b = c$.

⚙️ Process

Solve Any Simple Equation in 3 Steps

Isolate, simplify, verify.

↳ Undo operations in reverse order, then always verify by substitution.

⚠️ Common Mistake

Don't Change Only One Side

One-sided changes break the balance.

↳ Every operation must be mirrored on both sides — no exceptions.

⚠️ Common Mistake

Sign Errors When Transposing

Flip the sign when you cross over.

↳ Transposing a term across '=' reverses its sign.

✏️ Example

Worked Example: Variable on Both Sides

Bring variables together first.

↳ Collect all variable terms on one side, constants on the other.

🌍 Real World

Equations Solve Everyday Problems

From bills to budgets, equations rule.

↳ Any 'find the unknown' real-life problem often becomes a simple equation.

🧠 Memory Trick

Remember: Opposite Operations Undo

Addition undoes subtraction — always.

↳ Use the inverse operation to peel away everything around the variable.

🎯 Exam Tip

Score Full Marks: Show Every Step

Steps earn marks, not just answers.

↳ Examiners award method marks — never skip the working.

⚖️ Comparison

Expression vs Equation

One has '=', the other doesn't.

↳ An equation is two expressions joined by '='; an expression alone cannot be 'solved'.

⭐ Important Fact

A Simple Equation Has Exactly One Solution

One variable, one answer.

A linear equation in one variable ($ax + b = c$, $a \neq 0$) has exactly one solution — the value that makes both sides equal.

↳ Degree-1 equations in one variable always have a unique solution.

🛠️ Application

Setting Up an Equation from a Word Problem

Translate words into math.

↳ Word problems become easy once you translate them into an equation.

🚫 Warning

Don't Forget to Divide by the Coefficient

The number in front matters.

↳ Always divide by the coefficient to fully isolate the variable.

❓ Did You Know

The Word 'Algebra' Comes from Arabic

A 1,200-year-old math tradition.

The word *algebra* comes from al-jabr, a term used by the 9th-century mathematician Al-Khwarizmi in his book on solving equations.

↳ Equation-solving is one of the oldest tools in mathematics.

Smart notes

What you'll study, topic by topic

Simple Equations

15 min

Simple equations are algebraic statements where two expressions are set equal, containing one variable of degree one. Solving them means finding the value of the variable that makes the equation true, using inverse opera…

A **simple equation** is one where the unknown appears only to the first power, with no products of unknowns and no unknowns in denominators…
The general form of a simple equation in one variable is $ax + b = c$ with $a \neq 0$.
The **balance model** says: whatever operation you apply to one side, apply to the other.

Full explanation, worked examples & memory tricks in the app

Leveled MCQ practice

Try the smart MCQs from this kit

All 16 questions, laddered from warm-up to topper-level, each with its answer and the reasoning behind it.

Warm-up · 7 questions

Q1. Which of the following is a simple equation?

Beginner

A $x^2 + 3 = 7$

B $\frac{1}{x} + 2 = 5$

C $xy = 6$

D $2x + 5 = 11$

Show the answer

D · $2x + 5 = 11$

A simple equation has the unknown to the first power only, with no products of unknowns and no unknowns in denominators. $2x + 5 = 11$ fits this; the others involve a square, a denominator, or a product.

Q2. What is the general form of a simple equation in one variable?

Beginner

A $ax + b = c$ with $a \neq 0$

B $ax^2 + b = c$

C $axy + b = c$

D $\frac{a}{x} + b = c$

Show the answer

A · $ax + b = c$ with $a \neq 0$

The general form is $ax + b = c$ with $a \neq 0$, where the unknown $x$ appears only to the first power.

Q3. According to the balance model, what must you do when you add 3 to the left side of an equation?

Beginner

A Add 3 to the right side as well

B Subtract 3 from the right side

C Leave the right side unchanged

D Multiply the right side by 3

Show the answer

A · Add 3 to the right side as well

The balance model requires that whatever operation you apply to one side, you apply the same operation to the other side to keep the equation balanced.

Q4. To solve $3x + 4 = 19$, what is the first step?

Beginner

A Multiply both sides by 3

B Subtract 4 from both sides

C Add 4 to both sides

D Divide both sides by 3

Show the answer

B · Subtract 4 from both sides

Undo operations in reverse order: the last operation applied to $x$ was adding 4, so first subtract 4 from both sides to get $3x = 15$.

Q5. What is the solution to $2x + 5 = 11$?

Beginner

A $x = 2$

B $x = 3$

C $x = 5.5$

D $x = 8$

Show the answer

B · $x = 3$

Subtract 5 from both sides: $2x = 6$. Divide both sides by 2: $x = 3$. Check: $2(3) + 5 = 11$.

Q6. Which statement correctly distinguishes an expression from an equation?

Beginner

A An expression has no equals sign and is simplified; an equation has an equals sign and is solved.

B An expression is solved; an equation is simplified.

C An expression has an equals sign; an equation does not.

D There is no difference; both are solved the same way.

Show the answer

A · An expression has no equals sign and is simplified; an equation has an equals sign and is solved.

An expression (like $3x + 5$) has no equals sign and is simplified. An equation (like $3x + 5 = 20$) has an equals sign and is solved for the unknown.

Q7. In a simple (linear) equation in one variable, the unknown has degree exactly $1$, which is why it has exactly one solution.

Beginner

A True

B False

Show the answer

A · True

In a simple (linear) equation in one variable, the unknown has degree exactly $1$, which is why it has exactly one solution.

Building up · 6 questions

Q8. Solve $5x - 3 = 2x + 9$.

Intermediate

A $x = 4$

B $x = 6$

C $x = 2$

D $x = 12$

Show the answer

A · $x = 4$

Subtract $2x$ from both sides: $3x - 3 = 9$. Add 3 to both sides: $3x = 12$. Divide by 3: $x = 4$. Check: $5(4) - 3 = 17$ and $2(4) + 9 = 17$.

Q9. A student solves $4x + 6 = 2x + 14$ by first dividing both sides by 2, getting $2x + 3 = x + 7$. What is the main drawback of this approach?

Intermediate

A It introduces an extraneous solution.

B It makes the equation messier; collecting terms first is cleaner.

C It is mathematically invalid.

D It changes the solution of the equation.

Show the answer

B · It makes the equation messier; collecting terms first is cleaner.

Dividing by 2 is valid but often makes the equation messier. It is usually cleaner to first move all $x$-terms to one side and constants to the other, then divide.

Q10. In the equation $x + 7 = 12$, transposing 7 to the right side gives:

Intermediate

A $x = 7 - 12$

B $x = 12 - 7$

C $x = 12 \times 7$

D $x = 12 + 7$

Show the answer

B · $x = 12 - 7$

Transposition moves a term to the other side and flips its sign. Moving $+7$ to the right gives $-7$, so $x = 12 - 7 = 5$.

Q11. Translate 'seven more than twice a number is 19' into an equation.

Intermediate

A $7x + 2 = 19$

B $x + 7 = 19$

C $2(x + 7) = 19$

D $2x + 7 = 19$

Show the answer

D · $2x + 7 = 19$

'Twice a number' is $2x$; 'seven more than' adds 7; 'is' signals equals. So $2x + 7 = 19$.

Q12. A student solves $3x + 4 = 19$ and gets $x = 5$. Which check confirms the solution?

Intermediate

A Substitute $x = 5$ into $x = 5$

B Substitute $x = 5$ into $3x + 4$ only

C Substitute $x = 5$ into $3x = 15$

D Substitute $x = 5$ into the original equation $3x + 4 = 19$

Show the answer

D · Substitute $x = 5$ into the original equation $3x + 4 = 19$

A solution is correct only if it satisfies the original equation. Substituting $x = 5$ into $3x + 4 = 19$ gives $19 = 19$, confirming it.

Q13. Which of the following is NOT a simple equation?

Intermediate

A $4x - 1 = 7$

B $2x = 10$

C $x/2 + 3 = 5$

D $x^2 = 9$

Show the answer

D · $x^2 = 9$

$x^2 = 9$ has the unknown squared, so it is not a simple (linear) equation. The others all have the unknown to the first power only.

Topper level · 3 questions

Q14. Solve $4(x - 2) = 3x + 8$.

Advanced

A $x = 16$

B $x = 8$

C $x = 0$

D $x = 4$

Show the answer

A · $x = 16$

Expand the left side: $4x - 8 = 3x + 8$. Subtract $3x$ from both sides: $x - 8 = 8$. Add 8: $x = 16$. Check: $4(16 - 2) = 56$ and $3(16) + 8 = 56$.

Q15. In solving $5x - 3 = 2x + 9$, a student subtracts $2x$ from the left side only, getting $3x - 3 = 9$. What error did the student make?

Advanced

A They applied the operation to only one side, violating the balance model.

B They should have added $2x$ to both sides.

C They subtracted instead of adding.

D They should have divided by 2 first.

Show the answer

A · They applied the operation to only one side, violating the balance model.

The balance model requires the same operation on both sides. Subtracting $2x$ from only the left side breaks the balance; it must be subtracted from both sides.

Q16. Which equation has the unknown on both sides and requires collecting terms before solving?

Advanced

A $3x + 4 = 19$

B $5x - 3 = 2x + 9$

C $2x = 10$

D $x + 7 = 12$

Show the answer

B · $5x - 3 = 2x + 9$

In $5x - 3 = 2x + 9$, the unknown appears on both sides. You must collect $x$-terms on one side and constants on the other before dividing.

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