Linear Equations in One Variable ICSE Class 8: Methods and Word Problems

What Linear Equations in One Variable is about

Weightage: CISCE publishes no question paper, marking scheme or chapter weightage for Class VIII Mathematics: your school sets and marks the exam. Linear Equations in One Variable sits in the Algebra theme of the CISCE Curriculum for Upper Primary Classes (VI-VIII).Source: CISCE Curriculum for Upper Primary Classes (VI-VIII), RDCD 2016

Key concepts

Solving by balancing and transposing

An equation is like a balance. You keep it level by doing the same thing to both sides: add, subtract, multiply or divide (but never divide by 0).

Transposing is the shortcut:

• a term that is added on one side becomes subtracted on the other,
• a factor that multiplies one side divides the other.

Always check by putting the answer back into the original equation.

Worked example

Solve $3x+7=22$.

Try it first, then show the solutionSolution

1. Transpose 7: $3x=22-7=15$.
2. Divide by 3: $x=5$.
3. Check: $3(5)+7=22$ ✓.

Answer: $x=5$

Variable on both sides

Collect the variable terms on one side (usually the side with the larger coefficient, to keep it positive) and the numbers on the other. Then divide by the coefficient.

Worked example

Solve $5x-4=2x+11$.

Try it first, then show the solutionSolution

1. Transpose $2x$ to the left and $-4$ to the right: $5x-2x=11+4$.
2. $3x=15$, so $x=5$.
3. Check: $25-4=21$ and $10+11=21$ ✓.

Answer: $x=5$

Brackets and fractions

• Brackets: expand first, multiplying every term inside: $2(x-3)=2x-6$.
• Fractions: multiply every term on both sides by the LCM of the denominators. This clears all the fractions in one step.

Put brackets round a numerator with more than one term before multiplying: $12\times \frac{x+1}{3}=4(x+1)$.

Worked example

Solve $\frac{x}{3}+\frac{x}{4}=14$.

Try it first, then show the solutionSolution

1. LCM of 3 and 4 is 12. Multiply every term by 12: $4x+3x=168$.
2. $7x=168$, so $x=24$.
3. Check: $8+6=14$ ✓.

Answer: $x=24$

Number problems

Translate the words:

• "a number" → $x$; "twice a number" → $2x$; "a number exceeds another by 5" → $x$ and $x-5$ (or $x+5$);
• "consecutive numbers" → $x$, $x+1$, $x+2$; "consecutive even (or odd) numbers" → $x$, $x+2$, $x+4$;
• "is", "gives", "equals" → $=$.

Then solve and give all the numbers asked for.

Worked example

One number is 4 times another, and their difference is 27. Find the numbers.

Try it first, then show the solutionSolution

1. Let the smaller number be $x$; the larger is $4x$.
2. $4x-x=27\Rightarrow 3x=27\Rightarrow x=9$.
3. Numbers: 9 and 36. Check: $36-9=27$ ✓.

Answer: 9 and 36

Age problems

Make a small table of present ages and ages after (or before) some years. Everyone gets older by the same number of years.

	Now	In $n$ years
Son	$s$	$s+n$
Father	$3s$	$3s+n$

Then use the condition ("twice", "three times") on the correct column.

Worked example

A father is three times as old as his son. In 12 years he will be twice as old as his son. Find their present ages.

Try it first, then show the solutionSolution

1. Let the son be $s$; father $3s$. In 12 years: $s+12$ and $3s+12$.
2. $3s+12=2(s+12)=2s+24$.
3. $s=12$; father $=36$. Check: in 12 years 24 and 48, and $48=2\times 24$ ✓.

Answer: Son 12 years, father 36 years

Money, coins and measurement problems

• Coins/notes: if there are $x$ coins of one kind out of $N$, the rest are $N-x$. Value $=$ (number) × (face value).
• Rectangles: perimeter $=2(l+b)$; write $l$ in terms of $b$.
• Angles: angles of a triangle add to $180°$; angles on a straight line add to $180°$.
• Speed: time $=\frac{\text{distance}}{\text{speed}}$.

Write units next to your final answer.

Worked example

A box has 20 coins, some ₹5 coins and the rest ₹2 coins. The total value is ₹70. How many of each coin are there?

Try it first, then show the solutionSolution

1. Let the number of ₹5 coins be $x$; ₹2 coins $=20-x$.
2. $5x+2(20-x)=70\Rightarrow 3x+40=70$.
3. $x=10$. So 10 coins of ₹5 and 10 coins of ₹2. Check: $50+20=70$ ✓.

Answer: 10 coins of ₹5 and 10 coins of ₹2

Maths in real life: Linear Equations in One Variable

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.

Topics in this chapter

• Equations with Fractions and Variable on Both Sides
• Word Problems on Linear Equations

Linear Equations in One Variable formula sheet

General form

$ax+b=c$

Solution x = (c − b)/a, a ≠ 0.

Consecutive numbers

$x,x+1,x+2$

Even or odd: x, x + 2, x + 4.

Perimeter of a rectangle

$P=2(l+b)$

Write l in terms of b.

Time from speed

$t=\frac{d}{s}$

Journey problems.

Value of coins

$V=n\times v$

Number × face value.

How the school exam asks Linear Equations in One Variable

The question patterns to expect, and the method that works for each.

Tricks to remember Linear Equations in One Variable

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

• Cross the bridge, change the sign

  When a term crosses the = sign, + becomes −, and × becomes ÷.

  Use it when: Transposing quickly and correctly.

• LCM wipes out fractions

  Multiply every term by the LCM of the denominators and all the fractions disappear.

  Use it when: Equations with fractions.

• Let, form, solve, answer, check

  Five steps for every word problem: Let x, Form the equation, Solve, Answer in words, Check in the story.

  Use it when: Word problems.

• Everyone ages together

  In age problems, add the same number of years to every person.

  Use it when: Age tables.

Linear Equations in One Variable: exam-level solved questions

Ordered from exam level to HOTS. Cover the solution, write your own, then compare step by step.

Case-study practice (Section B style)

Read the situation, then answer every part. In a school exam set on the board pattern this is a 10-mark question split into 3 + 3 + 4 marks.

Where the marks are

How a typical answer is split into step marks. Show every step: a right final answer alone rarely earns full marks.

Where students lose marks in Linear Equations in One Variable

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

• ✗Mistake: Not changing the sign when transposing

  A term keeps its effect only if its sign changes as it crosses.

  Wrong working

  $3x+7=22\Rightarrow 3x=22+7$

  Right working

  $3x=22-7=15$

  ✓Do this: Say 'cross and change' every time.

  Usually costs: The answer.

• ✗Mistake: Multiplying only some terms by the LCM

  Every term on both sides must be multiplied.

  Wrong working

  $\frac{x}{3}+\frac{x}{4}=14\Rightarrow 4x+3x=14$

  Right working

  $4x+3x=168$

  ✓Do this: Draw an arrow from the LCM to every term.

  Usually costs: The answer.

• ✗Mistake: Expanding a bracket with a minus wrongly

  $-3(x-1)=-3x+3$, not $-3x-3$.

  ✓Do this: Multiply the sign as well as the number.

  Usually costs: 1 to 2 marks.

• ✗Mistake: Answering only x

  The question asks for the ages, coins or numbers, not just $x$.

  ✓Do this: Finish with a sentence that answers the question, with units.

  Usually costs: 1 mark.

• ✗Mistake: Adding years to only one person

  In age problems everyone gets older.

  Wrong working

  $5s+6=3s$

  Right working

  $5s+6=3(s+6)$

  ✓Do this: Use an age table with a 'later' column for everyone.

  Usually costs: The whole problem.

Practice questions on Linear Equations in One Variable

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

1. Q1 · 1 mark · MCQExam level

   If $3(x-2)=12$, then $x$ equals:
   (a) 2
   (b) 4
   (c) 6
   (d) 14

   Hint

   Divide by 3 first.

   Answer

   (c)

2. Q2 · 2 marks · Short answerExam level

   Solve $7x-9=4x+15$.

   Hint

   Collect x terms on the left.

   Answer

   $3x=24$, $x=8$.

3. Q3 · 2 marks · Short answerExam level

   A number is multiplied by 6 and then 5 is subtracted. The result is 31. Find the number.

   Hint

   $6x-5=31$.

   Answer

   $6x=36$, $x=6$.

4. Q4 · 3 marks · Short answerHarder exam level

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

   Hint

   Cross-multiply.

   Answer

   $3(2x-3)=4(x+1)\Rightarrow 6x-9=4x+4\Rightarrow x=\frac{13}{2}$.

5. Q5 · 1 mark · Assertion-reasonHarder exam level

   Assertion (A): $x=4$ is the solution of $3x-2=10$.
   Reason (R): Putting $x=4$ makes the left side equal to the right side.
   (a) (A) is true and (R) is false.
   (b) (A) is false and (R) is true.
   (c) Both (A) and (R) are true, and (R) is the correct explanation for (A).
   (d) Both (A) and (R) are true, but (R) is not the correct explanation for (A).

   Hint

   Substitute.

   Answer

   (c)

6. Q6 · 3 marks · Short answerHarder exam level

   The perimeter of a rectangle is 70 cm. Its length is 5 cm more than its breadth. Find the length and breadth.

   Hint

   $2(b+b+5)=70$.

   Answer

   $4b+10=70$, $b=15$ cm, $l=20$ cm.

7. Q7 · 3 marks · Short answerHarder exam level

   Divide ₹1,200 between A and B so that A gets ₹150 more than twice B's share.

   Hint

   B = x, A = 2x + 150.

   Answer

   $3x+150=1200$, $x=350$. B ₹350, A ₹850.

8. Q8 · 3 marks · Short answerHarder exam level

   In an isosceles triangle, the angle between the equal sides is 30° more than each base angle. Find all three angles.

   Hint

   Base angles $b$, $b$; vertex $b+30$.

   Answer

   $3b+30=180$, $b=50°$. Angles 50°, 50°, 80°.

9. Q9 · 4 marks · Long answerHarder exam level

   The present ages of A and B are in the ratio $5:7$. In 4 years, the sum of their ages will be 56. Find their present ages.

   Hint

   Ages $5k$ and $7k$.

   Answer

   $5k+4+7k+4=56\Rightarrow 12k=48\Rightarrow k=4$. A 20 years, B 28 years.

10. Q10 · 3 marks · Short answerHarder exam level

    When a number is divided by 4 and 5 is added, the result equals the number divided by 3. Find the number.

    Hint

    $\frac{x}{4}+5=\frac{x}{3}$.

    Answer

    Multiply by 12: $3x+60=4x$, $x=60$.

11. Q11 · 4 marks · Long answerHOTS / competency

    In a class, $\frac{1}{4}$ of the students are in the hall, $\frac{2}{5}$ of the rest are in the library, and the remaining 18 are in the classroom. How many students are in the class?

    Hint

    Rest after the hall is $\frac{3}{4}x$.

    Answer

    Classroom $=\frac{3}{5}\times \frac{3}{4}x=\frac{9}{20}x=18$, so $x=40$.

12. Q12 · 4 marks · Long answerHOTS / competency

    Asha travels 16 km in 2 hours. She walks part of the way at 4 km/h and cycles the rest at 12 km/h. How far does she walk?

    Hint

    Time walking + time cycling = 2.

    Answer

    $\frac{d}{4}+\frac{16-d}{12}=2\Rightarrow 3d+16-d=24\Rightarrow d=4$ km walked (1 h), 12 km cycled (1 h).

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. 1.What is a linear equation in one variable?Show answer

   An equation with one variable whose highest power is 1.

2. 2.Solve 2x = 18.Show answer

   x = 9.

3. 3.Solve x − 7 = 3.Show answer

   x = 10.

4. 4.Solve x/5 = 4.Show answer

   x = 20.

5. 5.Write three consecutive odd numbers starting at x.Show answer

   x, x + 2, x + 4.

6. 6.If a number is increased by 8 to give 23, what is the number?Show answer

   15.

7. 7.What do you multiply by to clear the fractions in x/6 + x/9 = 5?Show answer

   18 (the LCM).

8. 8.How do you check a solution?Show answer

   Substitute it into the original equation; both sides must be equal.

Quick revision checklist

The night before the exam: can you do each of these without looking?

• ☐Did you say what x stands for?
• ☐Did you change signs when transposing?
• ☐Did you multiply every term by the LCM?
• ☐Did you expand brackets with the correct signs?
• ☐Did you answer the actual question, with units?
• ☐Did you check the answer in the story?

Linear Equations in One Variable: frequently asked questions