MathsICSE Class 9 · Simultaneous Linear Equations

Cross-multiplication, Special Patterns and Word Problems (ICSE Class 9)

Updated 17 October 2026 · Fact-checked

For a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0, cross-multiplication gives x/(b₁c₂ − b₂c₁) = y/(c₁a₂ − c₂a₁) = 1/(a₁b₂ − a₂b₁). Swapped coefficients are solved by adding and subtracting; reciprocal equations by putting u = 1/x and v = 1/y. For word problems, define two variables, form two equations, solve and check against the story.

In each 2×2 block multiply along the blue arrow, subtract the orange product, then divide each top value by the bottom −13.

Illustration drawn for QuestionBuddy.

Understand Cross-multiplication, Special Patterns and Word Problems on Simultaneous Equations

Cross-multiplication. Write the equations as $a_{1} x + b_{1} y + c_{1} = 0$ and $a_{2} x + b_{2} y + c_{2} = 0$ . Then
$ $\frac{x}{b _{1} c _{2} - b _{2} c _{1}} = \frac{y}{c _{1} a _{2} - c _{2} a _{1}} = \frac{1}{a _{1} b _{2} - a _{2} b _{1}} .$ $
For $2 x + 3 y - 13 = 0$ and $3 x - 2 y = 0$ : $a_{1} = 2, b_{1} = 3, c_{1} = - 13$ and $a_{2} = 3, b_{2} = - 2, c_{2} = 0$ . Then $b_{1} c_{2} - b_{2} c_{1} = - 26$ , $c_{1} a_{2} - c_{2} a_{1} = - 39$ , $a_{1} b_{2} - a_{2} b_{1} = - 13$ , so $x = 2$ , $y = 3$ . The method needs $a_{1} b_{2} - a_{2} b_{1} \neq = 0$ .

Swapped coefficients. For $37 x + 43 y = 123$ and $43 x + 37 y = 117$ , add: $80 (x + y) = 240$ , $x + y = 3$ ; subtract: $- 6 x + 6 y = 6$ , $y - x = 1$ . So $x = 1$ , $y = 2$ .

Reciprocal equations. For $\frac{2}{x} + \frac{3}{y} = 7$ and $\frac{3}{x} - \frac{2}{y} = 4$ put $u = \frac{1}{x}$ , $v = \frac{1}{y}$ : $2 u + 3 v = 7$ and $3 u - 2 v = 4$ . Solving, $u = 2$ , $v = 1$ , so $x = \frac{1}{2}$ , $y = 1$ . Remember to convert back.

Word problems. Read twice; name the unknowns; translate each fact into an equation; solve; check against the words.

*Digits:* two-digit number with tens digit $x$ and units $y$ is $10 x + y$ ; reversed is $10 y + x$ .
*Ages:* 'n years ago' means subtract n from both ages; 'after n years' adds.
*Speed:* upstream speed $= u - v$ , downstream $= u + v$ .
*Money:* number of items $\times$ price.

Level. Class IX is a school-exam class. Reciprocal equations and age problems with two time frames are at the hardest school-exam level.

Cross-multiplication, Special Patterns and Word Problems on Simultaneous Equations: formulas to remember

Cross-multiplication: $\frac{x}{b _{1} c _{2} - b _{2} c _{1}} = \frac{y}{c _{1} a _{2} - c _{2} a _{1}} = \frac{1}{a _{1} b _{2} - a _{2} b _{1}}$; Equations written with =0.
Two-digit number: $10 x + y$; Reversed: 10y + x.
Boat speeds: $upstream u - v, downstream u + v$; u boat, v stream.
Reciprocal substitution: $u = \frac{1}{x}, v = \frac{1}{y}$; Makes the equations linear.

How to solve Cross-multiplication, Special Patterns and Word Problems on Simultaneous Equations questions

Define unknowns, build two equations, pick the pattern-fit method, solve, check against the story.

1Write down what $x$ and $y$ stand for, with units.
2Translate each statement into an equation.
3Use cross-multiplication, add-and-subtract for swapped coefficients, or reciprocal substitution as the pattern suggests.
4Solve, then convert back if you substituted.
5Check both equations and the conditions in the problem.
6Answer in a full sentence with units.

Cross-multiplication, Special Patterns and Word Problems on Simultaneous Equations 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.

Ticket sales

Suppose a hall sells 40 tickets for ₹1,600 in total, adult tickets at ₹50 and child tickets at ₹30.

How many of each were sold?

Try it first, then show the solutionSolution

$a + c = 40$ and $50 a + 30 c = 1600$ .
Substitute $c = 40 - a$ : $50 a + 1200 - 30 a = 1600$ , $20 a = 400$ , $a = 20$ , $c = 20$ .

Answer: 20 adult and 20 child tickets.

Memory hook: Count equation and money equation.

Boat in a river

Imagine a boat taking 3 hours to go 12 km upstream and 2 hours to return downstream.

Find the boat's speed in still water and the stream's speed.

Try it first, then show the solutionSolution

Upstream $u - v = 4$ , downstream $u + v = 6$ .
Add: $2 u = 10$ , $u = 5$ ; $v = 1$ .

Answer: Boat 5 km/h; stream 1 km/h.

Memory hook: Add and subtract the two speeds.

Tricks to remember Cross-multiplication, Special Patterns and Word Problems on Simultaneous Equations

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

Add then subtract
Swapped coefficients: add for the sum, subtract for the difference.
Use it when: 37x + 43y type.
Flip to reciprocal
Put u = 1/x, v = 1/y, then flip back at the end.
Use it when: Fraction-variable equations.
Digit value
Tens digit counts ten times.
Use it when: Two-digit problems.

Cross-multiplication, Special Patterns and Word Problems on Simultaneous Equations: solved examples

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

Example 1: A quick pair · 1 mark · MCQBoard level

The solution of $x + y = 3$ and $x - y = 1$ is:
(a) $(1, 2)$
(b) $(2, 1)$
(c) $(3, 0)$
(d) $(0, 3)$

Try it first, then show the solutionSolution

Add: $2 x = 4$ , $x = 2$ , $y = 1$ .

Answer: (b)

Example 2: Cross-multiplication · 3 marks · Short answerBoard level

Solve $2 x + 3 y - 13 = 0$ and $3 x - 2 y = 0$ by cross-multiplication.

Try it first, then show the solutionSolution

$a_{1} = 2, b_{1} = 3, c_{1} = - 13$ ; $a_{2} = 3, b_{2} = - 2, c_{2} = 0$ .
$\frac{x}{0 - 26} = \frac{y}{- 39 - 0} = \frac{1}{- 4 - 9}$ , i.e. $\frac{x}{- 26} = \frac{y}{- 39} = \frac{1}{- 13}$ .
$x = 2$ , $y = 3$ . Check: $4 + 9 = 13$ ; $6 - 6 = 0$ . ✓

Answer: x = 2, y = 3

Example 3: Swapped coefficients · 4 marks · Short answerHarder board level

Solve $37 x + 43 y = 123$ and $43 x + 37 y = 117$ .

Try it first, then show the solutionSolution

Add: $80 x + 80 y = 240$ , so $x + y = 3$ .
Subtract (first minus second): $- 6 x + 6 y = 6$ , so $y - x = 1$ .
Adding these: $2 y = 4$ , $y = 2$ , $x = 1$ .
Check: $37 + 86 = 123$ ; $43 + 74 = 117$ . ✓

Answer: x = 1, y = 2

Example 4: Reciprocals · 4 marks · Short answerHarder board level

Solve $\frac{2}{x} + \frac{3}{y} = 7$ and $\frac{3}{x} - \frac{2}{y} = 4$ .

Try it first, then show the solutionSolution

Let $u = \frac{1}{x}$ , $v = \frac{1}{y}$ : $2 u + 3 v = 7$ , $3 u - 2 v = 4$ .
Multiply by 2 and 3: $4 u + 6 v = 14$ , $9 u - 6 v = 12$ . Add: $13 u = 26$ , $u = 2$ ; then $v = 1$ .
$x = \frac{1}{2}$ , $y = 1$ . Check: $4 + 3 = 7$ ; $6 - 2 = 4$ . ✓

Answer: $x = \frac{1}{2}$ , $y = 1$

Example 5: A digit problem · 5 marks · Long answerHOTS / competency

The sum of the digits of a two-digit number is $9$ . If the digits are interchanged, the number increases by $27$ . Find the number.

Try it first, then show the solutionSolution

Let the number be $10 x + y$ ; $x + y = 9$ ... (1).
Reversed: $10 y + x$ . $(10 y + x) - (10 x + y) = 27$ , so $9 (y - x) = 27$ and $y - x = 3$ ... (2).
Add (1) and (2): $2 y = 12$ , $y = 6$ , $x = 3$ .
Number $= 36$ ; reversed $= 63$ ; $63 - 36 = 27$ . ✓

Answer: The number is 36.

Where students lose marks in Cross-multiplication, Special Patterns and Word Problems on Simultaneous Equations

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

Mistake: Forgetting to convert back
The answer must be in x and y.
Wrong working
$u = 2$ , $v = 1$ as the final answer.
Right working
$x = \frac{1}{2}$ , $y = 1$ .
Do this: Flip back.
Usually costs: Final marks.
Mistake: Wrong sign in cross-multiplication
The constants must be on the same side as the other terms.
Wrong working
Using $c_{1} = 13$ for $2 x + 3 y = 13$ .
Right working
$c_{1} = - 13$ in $2 x + 3 y - 13 = 0$ .
Do this: Write both equations with 0 on the right.
Usually costs: Whole answer.
Mistake: Reversing the digits wrongly
Tens digit counts ten times.
Wrong working
Number is $x + y$ .
Right working
$10 x + y$ .
Do this: Use place value.
Usually costs: Whole answer.

Exam tips

Class IX is examined in school papers; word problems carry 4 to 5 marks.
Always write "Let ... be ..." with units.
Check the answer against the words of the problem.
In cross-multiplication, rewrite with 0 on the right first.

Practice questions on Cross-multiplication, Special Patterns and Word Problems on Simultaneous Equations

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
Solve $x + 2 y = 8$ and $3 x - 2 y = 0$ :
(a) $(2, 3)$
(b) $(3, 2)$
(c) $(4, 2)$
(d) $(1, 3.5)$
Hint
Add the equations.
Answer
(a). $4 x = 8$ , $x = 2$ , $y = 3$ .
Q2 · 4 marks · Short answerHarder board level
Five years ago, a father was $7$ times as old as his son. Five years from now he will be $3$ times as old. Find their present ages.
Hint
F − 5 = 7(S − 5); F + 5 = 3(S + 5).
Answer
$F - 7 S = - 30$ and $F - 3 S = 10$ ; subtract: $4 S = 40$ , $S = 10$ , $F = 40$ . Check: $35 = 7 \times 5$ , $45 = 3 \times 15$ .
Q3 · 4 marks · Short answerHarder board level
A boat takes $3$ hours to go $12$ km upstream and $2$ hours to return. Find the speed of the boat in still water and of the stream.
Hint
u − v = 4, u + v = 6.
Answer
$u = 5$ km/h and $v = 1$ km/h.
Q4 · 1 mark · Assertion-reasonHarder board level
Assertion (A): The equations $2 x + 3 y = 5$ and $4 x + 6 y = 11$ have no solution.
Reason (R): $\frac{a _{1}}{a _{2}} = \frac{b _{1}}{b _{2}} \neq = \frac{c _{1}}{c _{2}}$ .
(a) Both (A) and (R) are true and (R) explains (A).
(b) Both are true but (R) does not explain (A).
(c) (A) is true, (R) is false.
(d) (A) is false, (R) is true.
Hint
$\frac{2}{4} = \frac{3}{6} \neq = \frac{5}{11}$ .
Answer
(a)
Q5 · 4 marks · Case-basedHOTS / competency
Case study: A hall sells $40$ tickets for ₹1,600 in total. An adult ticket is ₹50 and a child ticket is ₹30.
(a) Form two equations.
(b) Find the number of adult and child tickets.
(c) Find the money collected from adult tickets.
Hint
Substitute c = 40 − a.
Answer
(a) a + c = 40; 50a + 30c = 1600 (b) 20 adult, 20 child (c) ₹1,000

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.Value of a two-digit number with digits x, y?Show answer
10x + y.
2.Upstream speed?Show answer
Boat speed minus stream speed.
3.What does u = 1/x do?Show answer
Turns reciprocal equations into linear ones.
4.Condition for cross-multiplication?Show answer
a₁b₂ − a₂b₁ ≠ 0.

Cross-multiplication, Special Patterns and Word Problems on Simultaneous Equations: frequently asked questions

Is cross-multiplication compulsory?

It is one of the listed methods; any method with working is accepted.

What if the denominator a₁b₂ − a₂b₁ is zero?

The lines are parallel or coincide; there is no unique solution.

How do I check a word problem?

Test both equations and the conditions in the story.