One Formula Solves Any Pair of Equations Without Rearranging
Apply the cross-multiplication formula to any pair of linear equations, then frame equations from word problems on ages, fractions, digits, speed and the sides of a rectangle.
Is there a single formula that solves any pair of linear equations?
Substitution and elimination both need a decision before you can start. Which variable do I make the subject? Which equation do I multiply, and by what? Those decisions are where marks leak away.
Cross-multiplication removes the decisions. You write both equations in one standard shape, compute three numbers, and read off the answer.
Worked example. Solve and .
First put both into the shape :
So , , and , , . Now the three numbers:
-
-
-
Then and .
Check in both equations: and , as required.
No rearranging, no choosing a multiplier — just six coefficients fed into one pattern.
This page covers the second part of the ICSE Class 9 Mathematics chapter on simultaneous linear equations: the cross-multiplication method, and framing equations from word problems on numbers, ages, fractions, speed, digits and the sides of a figure.
Cross-multiplication removes the decisions. You write both equations in one standard shape, compute three numbers, and read off the answer.
Worked example. Solve and .
First put both into the shape :
So , , and , , . Now the three numbers:
-
-
-
Then and .
Check in both equations: and , as required.
No rearranging, no choosing a multiplier — just six coefficients fed into one pattern.
This page covers the second part of the ICSE Class 9 Mathematics chapter on simultaneous linear equations: the cross-multiplication method, and framing equations from word problems on numbers, ages, fractions, speed, digits and the sides of a figure.
Formula
What exactly is the cross-multiplication formula?
**Write both equations as , then use the single chain below.**
For and ,
Each denominator is a cross product of two columns, taken in a fixed cycle: for you cross the and columns, for the and columns, and for the last one the and columns. **The cycle , , is the whole memory load.
Worked example.** Solve and .
Standard shape: and .
- denominator:
- denominator:
- last denominator:
Check: and , as required.
The two traps, both about signs.
- The constant must be moved to the left first. Using instead of flips the sign of two denominators and gives a wrong pair
- The middle denominator is , not . It is the one term in the chain where the order reverses
And notice what the last denominator is. is exactly the quantity that decides how many solutions a pair has. If it is zero you cannot divide, and that is the algebra refusing to give a unique answer — the two lines are parallel or identical. The formula and the geometry agree, because they are the same statement.
For and ,
Each denominator is a cross product of two columns, taken in a fixed cycle: for you cross the and columns, for the and columns, and for the last one the and columns. **The cycle , , is the whole memory load.
Worked example.** Solve and .
Standard shape: and .
- denominator:
- denominator:
- last denominator:
Check: and , as required.
The two traps, both about signs.
- The constant must be moved to the left first. Using instead of flips the sign of two denominators and gives a wrong pair
- The middle denominator is , not . It is the one term in the chain where the order reverses
And notice what the last denominator is. is exactly the quantity that decides how many solutions a pair has. If it is zero you cannot divide, and that is the algebra refusing to give a unique answer — the two lines are parallel or identical. The formula and the geometry agree, because they are the same statement.
How do you turn a word problem about numbers or ages into two equations?
Name the two unknowns first, then write one equation for each sentence of information.
A word problem gives you exactly as many facts as you need. Two unknowns need two facts, and each fact is usually one clause in the question.
Worked example 1 — two numbers. The sum of two numbers is and their difference is . Find them.
Let the numbers be and with :
Adding gives , so and then .
Check: and , as required.
Worked example 2 — ages. A father is three times as old as his son. In ten more years he will be twice as old as his son. Find their present ages.
Let the father be years and the son years now:
Substituting the first into the second: , so and .
Check: is three times , and later is twice , as required.
Worked example 3 — a fraction. A fraction becomes when is added to its numerator and is subtracted from its denominator; it becomes when is added to its denominator alone. Find the fraction.
Let the fraction be :
Substituting, , so and . The fraction is .
Check: and , as required.
The one habit that makes ages questions safe is writing now next to your variables. Almost every wrong answer in this section comes from writing — ageing the father but not the son. Both people age by the same amount, so the bracket on the right is not optional.
A word problem gives you exactly as many facts as you need. Two unknowns need two facts, and each fact is usually one clause in the question.
Worked example 1 — two numbers. The sum of two numbers is and their difference is . Find them.
Let the numbers be and with :
Adding gives , so and then .
Check: and , as required.
Worked example 2 — ages. A father is three times as old as his son. In ten more years he will be twice as old as his son. Find their present ages.
Let the father be years and the son years now:
Substituting the first into the second: , so and .
Check: is three times , and later is twice , as required.
Worked example 3 — a fraction. A fraction becomes when is added to its numerator and is subtracted from its denominator; it becomes when is added to its denominator alone. Find the fraction.
Let the fraction be :
Substituting, , so and . The fraction is .
Check: and , as required.
The one habit that makes ages questions safe is writing now next to your variables. Almost every wrong answer in this section comes from writing — ageing the father but not the son. Both people age by the same amount, so the bracket on the right is not optional.
How do you frame equations for speed, streams and two travellers?
**Convert every sentence into speed, distance or time using , then let the two unknown speeds be and .
Two standard shapes cover almost every question in this section.
Shape one — a boat and a stream.** If the boat's own speed is km/h and the stream's is km/h, then downstream the stream helps and upstream it resists:
Worked example. A boat covers km downstream in hours and km upstream in hours. Find both speeds.
Downstream speed km/h and upstream speed km/h, so
Adding, , so km/h and km/h.
Check: downstream km/h covers km in hours, and upstream km/h covers km in hours, as required.
Shape two — two travellers. A cyclist and a walker start from two towns km apart. Moving towards each other they meet in hours; moving in the same direction the cyclist catches the walker in hours. Find both speeds.
Towards each other the gap closes at ; in the same direction it closes at :
So km/h and km/h.
Check: closing at km/h clears km in hours, and closing at km/h clears it in hours, as required.
Why the same-direction case takes longer is worth seeing clearly rather than memorising. The gap is the same km both times, but the closing speed drops from to km/h, so the time rises by the same factor of three. Relative speed is the single idea behind both lines, and it returns in Class 11 as the first step in one-dimensional kinematics.
Two standard shapes cover almost every question in this section.
Shape one — a boat and a stream.** If the boat's own speed is km/h and the stream's is km/h, then downstream the stream helps and upstream it resists:
Worked example. A boat covers km downstream in hours and km upstream in hours. Find both speeds.
Downstream speed km/h and upstream speed km/h, so
Adding, , so km/h and km/h.
Check: downstream km/h covers km in hours, and upstream km/h covers km in hours, as required.
Shape two — two travellers. A cyclist and a walker start from two towns km apart. Moving towards each other they meet in hours; moving in the same direction the cyclist catches the walker in hours. Find both speeds.
Towards each other the gap closes at ; in the same direction it closes at :
So km/h and km/h.
Check: closing at km/h clears km in hours, and closing at km/h clears it in hours, as required.
Why the same-direction case takes longer is worth seeing clearly rather than memorising. The gap is the same km both times, but the closing speed drops from to km/h, so the time rises by the same factor of three. Relative speed is the single idea behind both lines, and it returns in Class 11 as the first step in one-dimensional kinematics.
How do you handle two-digit numbers and the sides of a rectangle?
**A two-digit number with tens digit and units digit is , and reversing it gives .**
That one line is the whole trick. Students who write the number as are multiplying the digits instead of placing them.
Worked example 1 — a reversed number. The digits of a two-digit number add to . Reversing the digits increases the number by . Find the number.
Adding the two equations, , so and . The number is .
Check: , and , as required.
**Notice the that appeared on its own.** The difference between a two-digit number and its reverse is always , so it is always a multiple of . If a question ever says the reverse exceeds the number by , the question is impossible — and spotting that is faster than solving.
Worked example 2 — a rectangle. The area of a rectangle stays the same when the length is increased by m and the breadth decreased by m. It also stays the same when the length is decreased by m and the breadth increased by m. Find the dimensions.
Let the length be m and the breadth m, so the area is .
Adding the two results, , so m, and then , giving m.
Check: the area is m; the first change gives m and the second gives m, as required.
The step that makes this question easy is that appears on both sides and cancels. What looks like a quadratic collapses into a linear pair the moment you expand. Expand first and judge afterwards — several apparently hard questions in this chapter are linear in disguise.
That one line is the whole trick. Students who write the number as are multiplying the digits instead of placing them.
Worked example 1 — a reversed number. The digits of a two-digit number add to . Reversing the digits increases the number by . Find the number.
Adding the two equations, , so and . The number is .
Check: , and , as required.
**Notice the that appeared on its own.** The difference between a two-digit number and its reverse is always , so it is always a multiple of . If a question ever says the reverse exceeds the number by , the question is impossible — and spotting that is faster than solving.
Worked example 2 — a rectangle. The area of a rectangle stays the same when the length is increased by m and the breadth decreased by m. It also stays the same when the length is decreased by m and the breadth increased by m. Find the dimensions.
Let the length be m and the breadth m, so the area is .
Adding the two results, , so m, and then , giving m.
Check: the area is m; the first change gives m and the second gives m, as required.
The step that makes this question easy is that appears on both sides and cancels. What looks like a quadratic collapses into a linear pair the moment you expand. Expand first and judge afterwards — several apparently hard questions in this chapter are linear in disguise.
Exam tip
What is the safest way to lay out a word problem in the exam?
Write the let statement, both equations, the working and the answer sentence as four separate blocks. Examiners award method marks for the framing even when the arithmetic slips, and they cannot award what they cannot find.
- Name the variables with units. *Let the boat's speed be km/h and the stream's speed be km/h. Not just let speed be * — a bare variable costs you the mark when the answer needs units
- Write both equations before solving either. Most lost marks come from starting to solve after one equation, then inventing the second to fit
- Number them (1) and (2) and refer to them in the working. Adding (1) and (2) is a full sentence of proof for one line of algebra
- Substitute back into the original words, not into your own rearranged line. If you mis-copied a coefficient, only the original catches it
- Answer the question that was asked. If the question wants the number, write , not and . If it wants the fraction, write
- For cross-multiplication, write the standard form as a separate line with the constants moved across, and label underneath. That single line prevents nearly every sign error
One more check that costs five seconds. Ages must be positive, speeds must be positive, a digit must be a whole number from to , and a breadth cannot exceed a length that the question calls the length. If your answer breaks one of those, you have a sign error rather than a wrong method — look at the last subtraction you did.
- Name the variables with units. *Let the boat's speed be km/h and the stream's speed be km/h. Not just let speed be * — a bare variable costs you the mark when the answer needs units
- Write both equations before solving either. Most lost marks come from starting to solve after one equation, then inventing the second to fit
- Number them (1) and (2) and refer to them in the working. Adding (1) and (2) is a full sentence of proof for one line of algebra
- Substitute back into the original words, not into your own rearranged line. If you mis-copied a coefficient, only the original catches it
- Answer the question that was asked. If the question wants the number, write , not and . If it wants the fraction, write
- For cross-multiplication, write the standard form as a separate line with the constants moved across, and label underneath. That single line prevents nearly every sign error
One more check that costs five seconds. Ages must be positive, speeds must be positive, a digit must be a whole number from to , and a breadth cannot exceed a length that the question calls the length. If your answer breaks one of those, you have a sign error rather than a wrong method — look at the last subtraction you did.
Did you know
Why does the cross-multiplication formula look like a determinant?
Look again at the three denominators. Each is of the form first times last minus last times first on a pair of columns. That pattern has a name you meet in Class 11: it is a two-by-two determinant.
Write the coefficients as a grid and the whole method becomes one object:
- the denominator uses the and columns
- the denominator uses the and columns
- the third uses the and columns
So cross-multiplication is not a separate trick — it is your first determinant, written without the notation. In Class 11 the same three numbers get names, and the rule for solving a pair of equations with them is stated in one line for any number of variables.
And the vanishing denominator becomes a theorem. When the coefficient ratios are equal, the lines are parallel or coincident, and the unique solution disappears. In the determinant language of Class 11 that single number decides the fate of the whole system — and for three equations in three unknowns it still does.
The useful takeaway is structural. The two methods in Part 1 were procedures; this one is a formula, and a formula can be examined for what it does at its own boundary. That habit — asking what happens when a denominator hits zero — is most of what distinguishes a strong algebra answer from a merely correct one.
Write the coefficients as a grid and the whole method becomes one object:
- the denominator uses the and columns
- the denominator uses the and columns
- the third uses the and columns
So cross-multiplication is not a separate trick — it is your first determinant, written without the notation. In Class 11 the same three numbers get names, and the rule for solving a pair of equations with them is stated in one line for any number of variables.
And the vanishing denominator becomes a theorem. When the coefficient ratios are equal, the lines are parallel or coincident, and the unique solution disappears. In the determinant language of Class 11 that single number decides the fate of the whole system — and for three equations in three unknowns it still does.
The useful takeaway is structural. The two methods in Part 1 were procedures; this one is a formula, and a formula can be examined for what it does at its own boundary. That habit — asking what happens when a denominator hits zero — is most of what distinguishes a strong algebra answer from a merely correct one.
Exam relevance
How does framing equations from words feed into JEE preparation?
Treat this chapter as the foundation layer for two later chapters that are both examined in JEE Main.
Where it leads. The coefficient arithmetic here becomes Matrices and Determinants in Class 11 and 12, where the same is the determinant of the coefficient matrix, and the condition for a unique solution is that it is non-zero. JEE Main regularly asks for the value of a parameter that makes a system inconsistent or gives infinitely many solutions — which is precisely the boundary case you met in Part 1 and again in the formula above.
What the question types look like. At this level the framing skill matters more than the solving. In JEE the same skill appears inside longer problems: a word setup in Physics kinematics (two bodies, relative speed), a mixture or concentration setup in Chemistry, and parameter conditions in Algebra. Expect the algebra itself to be one line inside a larger question rather than the whole question.
The single trap that costs marks. Writing the equations in the wrong standard form. For consistency conditions the system must be in form with the constant moved across, and the sign of decides whether you conclude no solution or infinitely many. Students who keep the constant on the right get the ratio test backwards.
Board versus competitive emphasis. The ICSE paper rewards the full layout — let statement, both equations, substitution and a check. A competitive paper never sees your working, so the premium shifts to speed and to recognising the boundary case on sight. Practise both: the long form now, and a quick ratio test for consistency alongside it.
Where it leads. The coefficient arithmetic here becomes Matrices and Determinants in Class 11 and 12, where the same is the determinant of the coefficient matrix, and the condition for a unique solution is that it is non-zero. JEE Main regularly asks for the value of a parameter that makes a system inconsistent or gives infinitely many solutions — which is precisely the boundary case you met in Part 1 and again in the formula above.
What the question types look like. At this level the framing skill matters more than the solving. In JEE the same skill appears inside longer problems: a word setup in Physics kinematics (two bodies, relative speed), a mixture or concentration setup in Chemistry, and parameter conditions in Algebra. Expect the algebra itself to be one line inside a larger question rather than the whole question.
The single trap that costs marks. Writing the equations in the wrong standard form. For consistency conditions the system must be in form with the constant moved across, and the sign of decides whether you conclude no solution or infinitely many. Students who keep the constant on the right get the ratio test backwards.
Board versus competitive emphasis. The ICSE paper rewards the full layout — let statement, both equations, substitution and a check. A competitive paper never sees your working, so the premium shifts to speed and to recognising the boundary case on sight. Practise both: the long form now, and a quick ratio test for consistency alongside it.
Key takeaways
What should you be able to do before moving on from simultaneous equations?
The second half of this chapter turns a solving technique into a modelling technique.
- Cross-multiplication solves any pair written as using
- Move the constant across first, and remember the middle denominator reverses its order
- Word problems on numbers and ages need one equation per clause, with now written beside every age variable
- Speed problems reduce to and : a stream helps one way and resists the other, and two travellers close a gap at the sum or the difference of their speeds
- **A two-digit number is **, its reverse is , and their difference is always a multiple of
- Area-unchanged questions look quadratic and collapse to linear because cancels on both sides
- A zero last denominator means no unique solution — the same parallel-lines condition from Part 1
Every one of these ends the same way: substitute your pair back into the original sentences, not into your own algebra. Set yourself five mixed questions — one age, one fraction, one boat, one reversed number, one rectangle — and see which framing you reach for first and which one you still have to think about.
- Cross-multiplication solves any pair written as using
- Move the constant across first, and remember the middle denominator reverses its order
- Word problems on numbers and ages need one equation per clause, with now written beside every age variable
- Speed problems reduce to and : a stream helps one way and resists the other, and two travellers close a gap at the sum or the difference of their speeds
- **A two-digit number is **, its reverse is , and their difference is always a multiple of
- Area-unchanged questions look quadratic and collapse to linear because cancels on both sides
- A zero last denominator means no unique solution — the same parallel-lines condition from Part 1
Every one of these ends the same way: substitute your pair back into the original sentences, not into your own algebra. Set yourself five mixed questions — one age, one fraction, one boat, one reversed number, one rectangle — and see which framing you reach for first and which one you still have to think about.