Pair of Linear Equations in Two Variables · Class 10 Mathematics: quick revision, 69 bites

Two unknowns, two equations

One fact is never enough

'My father is 30 years older than me' fits many pairs of ages.

Takeaway. Two unknowns generally need two equations.

General form of a pair

The shape to recognise

Takeaway. A solution must satisfy BOTH equations.

Turning words into equations

Name the letters first

Takeaway. The 'Let' line carries its own mark.

A pair from a shop bill

Pencils and erasers

Takeaway. One equation per fact in the story.

Drawing a line in two steps

Zero for x, zero for y

Put $x = 0$ to get the point on the y-axis; put $y = 0$ to get the point on the x-axis.

Takeaway. Two points are enough — then extend the line.

The solution on the graph

Where they cross

Every point on a line solves that equation.

Takeaway. One shared point, one solution.

What a pair of lines can do

Three pictures, three answers

Takeaway. There is no fourth case.

Plotting a point the right way round

x first, always

$(6, 0)$ means 6 across and 0 up.

Takeaway. The first coordinate is x.

Verify in both equations

Check costs ten seconds

For $x = 6$, $y = 0$: $6 + 3(0) = 6$ ✓ and $2(6) - 3(0) = 12$ ✓

Takeaway. A solution that fails either equation is wrong.

Reading the answer back into words

Champa's pants and skirts

Takeaway. Finish in the words of the question.

Equivalent equations

The same line, written twice

$5x - 8y + 1 = 0$ and $3x - \frac{24}{5}y + \frac{3}{5} = 0$ are the same line — multiply the second by $\frac53$.

Takeaway. One equation in disguise is not a second fact.

Extend your lines

Stop short and lose the point

Join the two points and continue the line across the whole graph, or the crossing point may fall outside what you drew.

Takeaway. A line has no ends.

Use algebra for awkward answers

When graphs mislead

$7x - 15y = 2$ with $x + 2y = 3$ meets at $\left(\frac{49}{29}, \frac{19}{29}\right)$ — unreadable on squared paper.

Takeaway. Neat coordinates: graph. Otherwise: algebra.

Consistent, inconsistent, dependent

Three words examiners use

Takeaway. A dependent pair is always consistent.

Pairs outside the classroom

Two bills, one plan

A phone plan with a fixed charge plus a rate per GB gives one equation per month's bill; two months' bills find both numbers.

Takeaway. Every 'two unknowns' story is this chapter.

Show your two points

The table earns marks

In a graphical question, write the small table of values for each equation before plotting. It is part of the marking scheme.

Takeaway. Table, plot, cross, check.

Two lines cannot cross twice

Never twice

Two different straight lines meet at one point or not at all; if they share two points they are the same line.

Takeaway. That is why only three pictures exist.

Decide before you draw

The coefficients already know

Put both equations as $a_1x + b_1y + c_1 = 0$ and compare $\frac{a_1}{a_2}$, $\frac{b_1}{b_2}$ and $\frac{c_1}{c_2}$.

Takeaway. No graph paper needed.

One solution

Different at the first step

The constants do not matter in this case.

Takeaway. Unequal a-to-b ratios mean the lines must cross.

Infinitely many solutions

All three agree

Takeaway. One equation is a multiple of the other.

No solution

The odd one out is c

Takeaway. Same slope, different constant — a gap that never closes.

Standard form first

Signs travel with c

$2x + 3y = 9$ becomes $2x + 3y - 9 = 0$, so $c = -9$.

Takeaway. Move everything to one side, then read the coefficients.

Ratios, pictures and names

The whole topic on one card

Takeaway. Three rows, and nothing else can happen.

x − 2y = 0 with 3x + 4y − 20 = 0

NCERT's own example

$\frac{1}{3} \neq \frac{-2}{4}$, so the lines intersect: one solution.

Takeaway. One comparison was enough.

2x + 3y − 9 = 0 with 4x + 6y − 18 = 0

The same line twice

$\frac{2}{4} = \frac{3}{6} = \frac{-9}{-18} = \frac12$ — all three equal.

Takeaway. Infinitely many solutions.

x + 2y − 4 = 0 with 2x + 4y − 12 = 0

So close, and never meeting

$\frac12 = \frac24$ but $\frac{-4}{-12} = \frac13$.

Takeaway. Parallel lines: no solution.

Always check c

Two out of three is not enough

Equal a and b ratios alone cannot tell coincident from parallel.

Takeaway. Answering after two comparisons is the classic trap.

Finding a missing k

Condition first, then solve

Takeaway. The check is where marks are won or lost.

Infinitely many solutions

A worked k

Takeaway. Two ratios find k; the third confirms it.

Consistent, inconsistent, dependent

Three words examiners use

Takeaway. 'Dependent' never means 'no solution'.

When a coefficient is 0

Do not divide by zero

Use cross-products instead: $a_1b_2 = a_2b_1$ says the same as $\frac{a_1}{a_2} = \frac{b_1}{b_2}$.

Takeaway. Safe whatever the coefficients are.

Why the test works

Slope and position

Equal a-to-b ratios mean equal slopes, so the lines are parallel or identical.

Takeaway. Slope first, position second.

Parallel lines in a bill

Two plans that never agree

Two mobile plans with the same rate per GB but different fixed charges give parallel lines — the bills are never equal.

Takeaway. Same slope, different constant.

How the marks are given

Write the ratios down

Show $\frac{a_1}{a_2}$, $\frac{b_1}{b_2}$ and $\frac{c_1}{c_2}$ with their values, then state the case and the name.

Takeaway. The comparison itself carries marks.

What substitution does

Two unknowns become one

Use one equation to write x in terms of y, then put that into the other equation.

Takeaway. Replace, then solve.

The method

Three steps, in order

Takeaway. The word OTHER is what makes it work.

Never substitute back into the same equation

The wasted page

From $x + 2y = 3$ take $x = 3 - 2y$. Putting it back gives $3 = 3$.

Takeaway. Rearrange one, substitute into the other.

Choosing the easiest variable

Hunt for the lonely 1

Pick a variable whose coefficient is 1 or −1 — it rearranges without fractions.

Takeaway. A good choice saves half the working.

7x − 15y = 2 with x + 2y = 3

NCERT's own example

Takeaway. Fractions in the answer are normal here.

Multiply the whole expression

Brackets or marks

$7(3 - 2y) = 21 - 14y$.

Takeaway. Put the bracket in before you expand.

A true statement

When both letters vanish

$18 = 18$ means the two equations say the same thing.

Takeaway. True statement, infinitely many solutions.

A false statement

The other degenerate case

$18 = 20$ can never hold, so no pair (x, y) works.

Takeaway. False statement, no solution.

What the statement tells you

Reading the leftover line

Takeaway. The leftover line names the case.

An age problem by substitution

Aftab and his daughter

$x - 7y = -42$ and $x - 3y = 6$ give $x = 6 + 3y$.

Takeaway. Ages 42 and 12 — and both conditions check.

Two-digit number problems

Digits are worth their place

Tens digit x and units digit y give the number $10x + y$, reversed $10y + x$.

Takeaway. Always write the number as 10x + y.

Rearranging safely

Sign of the moved term

From $x + 2y = 3$: $x = 3 - 2y$.

Takeaway. Move it, flip its sign.

How to verify

Both equations, every time

Put your final x and y into the ORIGINAL equations, not the rearranged one — the rearranged one cannot fail.

Takeaway. A check against your own working proves nothing.

Expect awkward answers

Fractions are not failure

$x = \frac{49}{29}$ is a perfectly good answer; keep fractions exact rather than rounding.

Takeaway. Round only if the question asks.

Do what the question says

The named method

If it says 'solve by substitution', elimination will not earn the method marks, however right the answer.

Takeaway. Read the instruction before starting.

Substitution or elimination

Two roads, one destination

Both methods always give the same solution; pick the one that suits the numbers.

Takeaway. Coefficient 1 → substitution is usually faster.

Where it comes from

An old method

Linear problems of this kind were solved in India by Brahmagupta in the 7th century.

Takeaway. The idea is over a thousand years old.

What elimination does

Make it vanish

Match the coefficient of one variable in both equations, then add or subtract to cancel it.

Takeaway. Match, then cancel.

The method

Three steps

Takeaway. Match, cancel, solve, back.

The only decision

Same Sign → Subtract

Both terms $+6y$? Subtract. One $+6y$ and one $-6y$? Add.

Takeaway. Two S words go together.

Including the constant

Multiply EVERY term

$3 \times (2x + 3y = 13)$ is $6x + 9y = 39$.

Takeaway. The constant is part of the equation.

2x + 3y = 13 and 5x − 2y = 4

A full solution

Takeaway. The LCM of 3 and 2 chose the multipliers.

Which variable to remove

Chase the smaller LCM

Either works. Eliminate the variable whose coefficients have the smaller LCM and the numbers stay small.

Takeaway. Same answer, less arithmetic.

Infinitely many

0 = 0 at the end

Both variables gone and a true statement left means the equations are equivalent — infinitely many solutions.

Takeaway. True statement, coincident lines.

No solution

0 = 2 at the end

A false statement means no pair can satisfy both — the lines are parallel.

Takeaway. False statement, parallel lines.

Substitution or elimination

Which method when

Takeaway. Both always give the same answer.

NCERT's 9 : 7 problem

Incomes in a ratio

Incomes ₹9x and ₹7x, spendings ₹4y and ₹3y, each saving ₹2000.

Takeaway. Incomes ₹18,000 and ₹14,000.

How a word problem is marked

Three marks, three parts

Takeaway. Each part earns its own mark.

Two-digit numbers

Place value in equations

Takeaway. Never write the number as x + y.

Age problems

Before and after

'Five years ago' subtracts 5 from BOTH ages; 'in ten years' adds 10 to both.

Takeaway. Change both people's ages together.

Boats and streams

Rowing with and against

Takeaway. Then time = distance ÷ speed.

The last mark

Finish in words

'The rectangle is 10 cm by 8 cm' earns the final mark; 'x = 10, y = 8' does not.

Takeaway. Answer the question that was asked.

Every sign flips

Subtracting a whole equation

$(5x - 2y = 4) - (5x + 3y = 13)$ gives $-5y = -9$.

Takeaway. Bracket the equation before subtracting.

It keeps the solutions

Why multiplying is allowed

Multiplying every term by a non-zero number gives an equivalent equation — the same line, so the same solutions.

Takeaway. That is what makes Step 1 legal.

Counting boards

An ancient method

Chinese mathematicians eliminated variables on counting boards around 200 BCE, in the Nine Chapters.

Takeaway. Computers still use the same idea.