Class 10 Maths Notes Chapter 3 — Pair of Linear Equations in Two Variables#

Key terms#

Linear equation in two variables — $ax + by + c = 0$, where $a$, $b$, $c$ are real numbers and $a$, $b$ are not both zero. Its graph is a straight line, and it has infinitely many solutions.
Pair of linear equations — two such equations in the same variables: $$a_1x + b_1y + c_1 = 0$$ $$a_2x + b_2y + c_2 = 0$$
Solution of the pair — a pair of values $(x, y)$ that satisfies both equations.
Consistent — has at least one solution. Inconsistent — has no solution.
Dependent — the two equations represent the same line (infinitely many solutions).

Graphical method and consistency#

Draw both lines on the same axes. Their position tells you the number of solutions, and so do the ratios of the coefficients:

The three ways two lines can lie, and the number of solutions each gives.

Ratios	Lines	Solutions	Pair is
$\dfrac{a_1}{a_2} \neq \dfrac{b_1}{b_2}$	Intersecting	Exactly one (unique)	Consistent
$\dfrac{a_1}{a_2} = \dfrac{b_1}{b_2} = \dfrac{c_1}{c_2}$	Coincident	Infinitely many	Dependent (consistent)
$\dfrac{a_1}{a_2} = \dfrac{b_1}{b_2} \neq \dfrac{c_1}{c_2}$	Parallel	None	Inconsistent

When the lines intersect, the point of intersection is the solution.
When they coincide, every point on the line is a solution.

Substitution method#

From one equation, write one variable in terms of the other, such as $y$ in terms of $x$.
Substitute it in the other equation to get an equation in one variable, and solve it.
Substitute this value back to find the other variable.

If step 2 gives a statement that is always true (like $9 = 9$), there are infinitely many solutions. If it gives a false statement (like $0 = 5$), there is no solution.

Elimination method#

Multiply the equations by suitable non-zero numbers so that one variable has equal (or opposite) coefficients.
Subtract (or add) the equations to eliminate that variable, and solve.
Substitute this value in either equation to find the other variable.

Example: $x + y = 14$ and $x - y = 4$.

Adding the two equations:

$$\begin{array}{rcl} x + y &=& 14 \\ x - y &=& 4 \\ \hline 2x &=& 18 \end{array}$$

$$\Rightarrow x = 9$$

Substituting $x = 9$ in $x + y = 14$:

$$9 + y = 14$$

$$\Rightarrow y = 5$$

Word problems#

Choose the variables clearly (“Let the cost of one pen be ₹$x$”).
Form one equation from each condition in the question.
Some equations are not linear at first (for example, with $\frac{1}{x}$ and $\frac{1}{y}$). They can be changed into a pair of linear equations by a substitution such as $u = \frac{1}{x}$, $v = \frac{1}{y}$.
Check the answer against the original conditions.

Common mistakes#

Comparing ratios before writing both equations in the form $ax + by + c = 0$. Signs of $c$ change when you move terms across.

Saying “consistent” only for a unique solution. Coincident lines (infinitely many solutions) are also consistent.

Forgetting to multiply every term, including the constant, in the elimination method.

Stopping after finding one variable. Always find both and check them in both equations.

3. Pair of Linear Equations