MathsICSE Class 8 · Linear Equations in One Variable

Linear Equations with Fractions and Variables on Both Sides (ICSE Class 8 Maths)

Updated 17 October 2026 · Fact-checked

Remove brackets, clear fractions by multiplying both sides by the LCM of the denominators, collect the variable terms on one side and the numbers on the other, then divide by the coefficient. If an equation has the form $\frac{a}{b} = \frac{c}{d}$, cross-multiply. Always check by substitution.

Notice each step: multiply every term by the LCM, expand brackets with signs, collect terms, then check x = 13 in the original equation.

Illustration drawn for QuestionBuddy.

Understand Equations with Fractions and Variable on Both Sides

The CISCE curriculum asks you to solve linear equations in one variable, avoiding complex coefficients.

Balance rule. Whatever you do to one side you must do to the other. Adding, subtracting, multiplying or dividing both sides by the same non-zero number keeps the equation true.

Transposing. Moving a term across the equals sign changes its sign: $5 x - 3 = 2 x + 9 \Rightarrow 5 x - 2 x = 9 + 3 \Rightarrow 3 x = 12$ .

Brackets. Expand first, minding signs: $3 (2 x - 1) - 2 (x + 3) = 11 \Rightarrow 6 x - 3 - 2 x - 6 = 11$ .

Fractions. Multiply every term by the LCM of the denominators: $\frac{x + 2}{3} - \frac{x - 1}{4} = 2$ with LCM $12$ gives $4 (x + 2) - 3 (x - 1) = 24$ . A fraction line acts as a bracket around its numerator.

Cross multiplication. For $\frac{3 x - 2}{5 x + 1} = \frac{1}{2}$ , multiply across: $2 (3 x - 2) = 5 x + 1$ .

Unknown constant. If $x = 3$ is a solution of $\frac{k x - 4}{x + 1} = 2$ , substitute $x = 3$ to get $\frac{3 k - 4}{4} = 2$ , so $k = 4$ .

Checking. Substitute the answer in the original equation; both sides must be equal. Do not check in a later, already-changed line.

Equations with Fractions and Variable on Both Sides: formulas to remember

Balance rule: $a = b \Rightarrow a \pm c = b \pm c, a c = b c$; Same operation on both sides.
Cross multiplication: $\frac{a}{b} = \frac{c}{d} \Rightarrow a d = b c$; For single fractions on both sides.
Clearing fractions: $multiply every term by the LCM$; Do not miss any term, including whole numbers.

How to solve Equations with Fractions and Variable on Both Sides questions

Clear brackets and fractions, collect like terms, then divide.

1Expand brackets with correct signs.
2Multiply by the LCM to clear fractions (every term).
3Collect $x$ terms on one side and constants on the other.
4Divide by the coefficient of $x$ .
5Substitute back to verify.

Equations with Fractions and Variable on Both Sides in real life

Each concept, met in an everyday situation. Read the situation, try the problem, then check the solution. The one-line hook is what to remember.

Equal charges

Suppose two phone plans charge Rs 200 plus Rs 3 per minute, and Rs 110 plus Rs 5 per minute.

For how many minutes are the two charges equal?

Try it first, then show the solutionSolution

$200 + 3 m = 110 + 5 m$ .
$90 = 2 m$ , so $m = 45$ .

Answer: 45 minutes (both cost Rs 335).

Memory hook: Equal costs make an equation with m on both sides.

Tricks to remember Equations with Fractions and Variable on Both Sides

Short memory aids. Say each one aloud once, then use it on a question the same day.

LCM for every term
Multiply every term by the LCM, including those with no fraction.
Use it when: Fractions in equations.
Transposing flips sign
A term changes sign when it crosses the equals sign.
Use it when: Collecting terms.

Equations with Fractions and Variable on Both Sides: solved examples

Ordered from board level to HOTS. Try each one before you open the solution.

Example 1: Variable on both sides · 1 mark · MCQBoard level

The solution of $5 x - 3 = 2 x + 9$ is:
(a) $x = 2$ (b) $x = 4$ (c) $x = 6$ (d) $x = 12$

Try it first, then show the solutionSolution

$5 x - 2 x = 9 + 3 \Rightarrow 3 x = 12 \Rightarrow x = 4$ .

Answer: (b)

Example 2: Brackets · 3 marks · Short answerBoard level

Solve $3 (2 x - 1) - 2 (x + 3) = 11$ .

Try it first, then show the solutionSolution

$6 x - 3 - 2 x - 6 = 11$ .
$4 x - 9 = 11 \Rightarrow 4 x = 20 \Rightarrow x = 5$ .
Check: $3 (9) - 2 (8) = 27 - 16 = 11$ .

Answer: $x = 5$

Example 3: Fractions · 3 marks · Short answerHarder board level

Solve $\frac{x + 2}{3} - \frac{x - 1}{4} = 2$ .

Try it first, then show the solutionSolution

Multiply by $12$ : $4 (x + 2) - 3 (x - 1) = 24$ .
$4 x + 8 - 3 x + 3 = 24 \Rightarrow x + 11 = 24 \Rightarrow x = 13$ .
Check: $\frac{15}{3} - \frac{12}{4} = 5 - 3 = 2$ .

Answer: $x = 13$

Example 4: Cross multiplication · 4 marks · Short answerHarder board level

Solve $\frac{3 x - 2}{5 x + 1} = \frac{1}{2}$ .

Try it first, then show the solutionSolution

Cross-multiply: $2 (3 x - 2) = 5 x + 1$ .
$6 x - 4 = 5 x + 1 \Rightarrow x = 5$ .
Check: $\frac{13}{26} = \frac{1}{2}$ .

Answer: $x = 5$

Example 5: Finding the constant · 4 marks · Long answerHOTS / competency

$x = 3$ is a solution of $\frac{k x - 4}{x + 1} = 2$ . Find $k$ and verify by solving the equation for that $k$ .

Try it first, then show the solutionSolution

Substitute $x = 3$ : $\frac{3 k - 4}{4} = 2 \Rightarrow 3 k - 4 = 8 \Rightarrow k = 4$ .
Equation becomes $\frac{4 x - 4}{x + 1} = 2 \Rightarrow 4 x - 4 = 2 x + 2$ .
$2 x = 6 \Rightarrow x = 3$ . Consistent.

Answer: $k = 4$

Where students lose marks in Equations with Fractions and Variable on Both Sides

The slips examiners see most often, and what to write instead.

Mistake: Not multiplying the whole-number term by the LCM
Every term, not just the fractions, must be multiplied.
Wrong working
$\frac{x + 2}{3} - \frac{x - 1}{4} = 2 \Rightarrow 4 (x + 2) - 3 (x - 1) = 2$ .
Right working
$\dots = 24$ .
Do this: Circle every term before multiplying.
Usually costs: Answer mark.
Mistake: Sign error when removing a bracket after a minus
The minus changes both signs inside.
Wrong working
$- 2 (x + 3) = - 2 x + 6$ .
Right working
$- 2 x - 6$ .
Do this: Multiply the factor into each term.
Usually costs: Answer mark.

Exam tips

Check the answer in the original equation.
Show the LCM clearly.
Write each step on a new line with the same sign of equality.

Practice questions on Equations with Fractions and Variable on Both Sides

Write each answer on paper first. Open the hint only if you are stuck, and the answer only when you have finished.

Q1 · 1 mark · MCQBoard level
The solution of $\frac{x}{3} + 2 = 7$ is:
(a) $x = 3$ (b) $x = 9$ (c) $x = 15$ (d) $x = 27$
Hint
Subtract 2, then multiply by 3.
Answer
(c)
Q2 · 2 marks · Short answerBoard level
Solve $9 x - 7 = 4 x + 13$ .
Hint
Collect x terms on the left.
Answer
$5 x = 20 \Rightarrow x = 4$ .
Q3 · 3 marks · Short answerHarder board level
Solve $\frac{2 x - 3}{4} + \frac{x + 1}{3} = \frac{25}{12}$ .
Hint
LCM is 12.
Answer
$3 (2 x - 3) + 4 (x + 1) = 25 \Rightarrow 6 x - 9 + 4 x + 4 = 25 \Rightarrow 10 x = 30 \Rightarrow x = 3$ .
Q4 · 3 marks · Short answerHarder board level
Solve $3 (x - 4) - 2 (2 x - 5) = 4$ .
Hint
Careful with the minus before the second bracket.
Answer
$3 x - 12 - 4 x + 10 = 4 \Rightarrow - x - 2 = 4 \Rightarrow x = - 6$ . Check: $3 (- 10) - 2 (- 17) = - 30 + 34 = 4$ .
Q5 · 4 marks · Long answerHOTS / competency
Solve $\frac{x - 1}{2} + \frac{x - 2}{3} + \frac{x - 3}{4} = 10$ .
Hint
Multiply every term by 12.
Answer
$6 (x - 1) + 4 (x - 2) + 3 (x - 3) = 120 \Rightarrow 13 x - 23 = 120 \Rightarrow x = 11$ . Check: $5 + 3 + 2 = 10$ .

Test your recall

Answer in your head or on paper first, then open the card. Pulling an answer out of memory makes it stick far better than re-reading.

1.What happens to a term when moved across the equals sign?Show answer
Its sign changes.
2.How do you clear fractions?Show answer
Multiply every term by the LCM of the denominators.
3.Cross-multiplication of a/b = c/d?Show answer
ad = bc.
4.Solve 2x + 5 = 17.Show answer
x = 6.

Linear Equations with Fractions and Variables on Both Sides (ICSE Class 8 Maths)

Understand Equations with Fractions and Variable on Both Sides

Equations with Fractions and Variable on Both Sides: formulas to remember

How to solve Equations with Fractions and Variable on Both Sides questions

Equations with Fractions and Variable on Both Sides in real life

Tricks to remember Equations with Fractions and Variable on Both Sides

Equations with Fractions and Variable on Both Sides: solved examples

Where students lose marks in Equations with Fractions and Variable on Both Sides

Exam tips

Practice questions on Equations with Fractions and Variable on Both Sides

Test your recall

Equations with Fractions and Variable on Both Sides: frequently asked questions

Can the answer be a fraction or negative?

What if both sides reduce to the same expression?