Linear Equations

In one market 2 pens and 1 notebook cost ₹30, and 1 pen and 2 notebooks cost ₹39. What is the price of each? We have two unknowns (the two prices) and two pieces of information — so we can find both. That is a system of linear equations. Linear means each unknown appears only to the first power — no $x^{2}$ or $x y$ — and the graph of such an equation is a straight line. Two lines can meet at one point (one solution), be parallel (no solution) or lie on each other (infinitely many).

a_{1} x + b_{1} y + c_{1} = 0, a_{2} x + b_{2} y + c_{2} = 0

A system in two variables

1

Equations in one variable

The balance rule

Keep the equation balanced like a scale and isolate $x$ .

Rules for solving

The same on both sides

Whatever you do, do it to both sides: add, subtract, multiply or divide (never by zero).

Open brackets, and clear fractions by multiplying by the LCM.
Bring the $x$ terms to one side and numbers to the other — a term changes sign when it crosses over.
Divide both sides by the coefficient of $x$ .
Check: put the value back into the original equation and see both sides agree.

Example 1 · Basic

Solve:

7 x - 3 = 4 x + 12

.

Solution, step by step

1Gather the $x$ terms on the left: $7 x - 4 x = 12 + 3$ .
2 $3 x = 15$ ⟹ $x = 5$ (check: $35 - 3 = 32 = 20 + 12$ ✓)

Answer $x = 5$

BasicTry it yourself first

Solve the equation

5 x + 3 = 2 x + 18

. Find

x

.

BasicTry it yourself first

Solve the equation

7 x - 4 = 3 x + 12

. Find

x

.

BasicTry it yourself first

Solve the equation

6 x + 5 = 4 x - 1

. Find

x

.

StandardTry it yourself first

Solve the equation

9 x + 8 = 4 x + 28

. Find

x

.

StandardTry it yourself first

Solve the equation

8 x - 7 = 3 x + 18

. Find

x

.

StandardTry it yourself first

Solve the equation

11 x + 2 = 6 x - 23

. Find

x

.

AdvancedTry it yourself first

Solve the equation

13 x - 9 = 5 x + 31

. Find

x

.

AdvancedTry it yourself first

Solve the equation

12 x + 14 = 7 x - 16

. Find

x

.

2

Two variables: three methods

Remove one variable

Finding $x$ and $y$ by substitution, elimination and cross multiplication.

Substitution method

Write one variable in terms of the other

Make $y = \dots$ from one equation and put it into the other.

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Elimination method

Equalise a coefficient and subtract

Multiply the equations by suitable numbers to make one coefficient equal.

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Cross multiplication

A direct formula

For $a_{1} x + b_{1} y + c_{1} = 0$ , $a_{2} x + b_{2} y + c_{2} = 0$ get $x$ and $y$ straight from the formula.

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3

Consistency and $k$ questions

Look at the ratios

Whether a system has one, no or infinitely many solutions — from the ratios $\frac{a _{1}}{a _{2}}, \frac{b _{1}}{b _{2}}, \frac{c _{1}}{c _{2}}$ .

The three cases

Lines: crossing, parallel, identical

Decide without solving by comparing the ratios.

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4

Word problems

From sentences to equations

Choose two unknowns, form an equation from each sentence, then solve.

Prices, animals and marks

Heads and feet, correct and wrong

Price pairs, heads and feet, exam marks and taxi fares.

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Ratios, money and numbers

“Give me” and savings

Coin ratios, exchanging money, income-expenditure and two numbers.

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5

Evaluating expressions and special forms

Simplify after solving

Values like $x^{2} + y^{2}$ , $(2 x + y)^{2}$ from the solution, exponent equations and three-variable sums.

Exponents, three variables and expressions

Identities and the adding trick

Find the unknowns first, then substitute; or use an identity for the expression directly.

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6

Traps and fast tricks

Slips · tricks

Where marks go in linear-equation questions.

The commonest mistakes

Six traps

The slips that cost marks.

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Fast tricks

Add-subtract and ratios

Habits for speed.

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7

The practice ladder

Basic → standard → advanced

New questions of every type, in three levels.

Level 1 — Basic

Direct rules

One-rule questions.

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Level 2 — Standard

Two ideas combined

Exam-level questions.

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Level 3 — Advanced

Many steps and clever thinking

Questions at the hard end of competitive exams.

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8

Revision — formula cards

Flip each card and test yourself

Read the front, think, then flip.

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Equations in one variable

Rules for solving

Two variables: three methods

Substitution method

Elimination method

Cross multiplication

Consistency and $k$ questions

The three cases

Word problems

Prices, animals and marks

Ratios, money and numbers

Evaluating expressions and special forms

Exponents, three variables and expressions

Traps and fast tricks

The commonest mistakes

Fast tricks

The practice ladder

Level 1 — Basic

Level 2 — Standard

Level 3 — Advanced

Revision — formula cards

Free All Access Passevery mind map · Hindi + English · 12 months

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