Class 9 Maths Chapter 13: Two Variables, One Line — Notes, Examples & Practice

A linear equation in two variables can be written as $ax + by + c = 0$, where $a$ and $b$ are not both zero. Examples: $2x + 3y = 12$, $y = 5x - 1$, $x = 4$ (here $b = 0$).

A solution is an ordered pair $(x, y)$ that makes the equation true. For $2x + 3y = 12$: $(0, 4)$, $(6, 0)$ and $(3, 2)$ are solutions. Order matters — $(2, 3)$ gives $4 + 9 = 13 \neq 12$, so it is not a solution.

There are infinitely many solutions: choose any $x$ and solve for $y$. In real situations only some may make sense — for example, the number of tickets must be a whole number.

Plot several solutions of a linear equation and they all lie on one straight line; every point on the line is a solution. To draw it, find two points (intercepts are easiest) and a third as a check.

Rearranged as $y = mx + d$, the number $m$ is the slope — how much $y$ changes when $x$ goes up by 1 — and $d$ is the y-intercept. From $ax + by + c = 0$ with $b \neq 0$: $m = -\dfrac{a}{b}$, $d = -\dfrac{c}{b}$.

Special lines: $y = k$ is horizontal (slope 0); $x = k$ is vertical and cannot be written as $y = mx + d$. The x-axis is $y = 0$ and the y-axis is $x = 0$.

A pair (system) of linear equations asks for $(x, y)$ satisfying both. On a graph, that is a point on both lines.

Two lines in a plane can: intersect at one point → exactly one solution (consistent); be parallel → no solution (inconsistent); or coincide → infinitely many solutions (dependent).

You can tell which without drawing by comparing the ratios of coefficients for $a_1x + b_1y + c_1 = 0$ and $a_2x + b_2y + c_2 = 0$.

Make one variable the subject of one equation, substitute that expression into the other equation, and solve the resulting one-variable equation. Then substitute back to get the second variable.

Choose the variable with coefficient 1 or −1 if possible — it avoids fractions.

Always check the answer in both original equations.

Multiply one or both equations by suitable numbers so that one variable has equal (or opposite) coefficients, then subtract (or add) the equations to eliminate it.

If eliminating one variable also eliminates the other and you get a false statement like $0 = 5$, the lines are parallel (no solution). A true statement like $0 = 0$ means the same line (infinitely many).

Step 1: name the two unknowns clearly (with units). Step 2: translate each sentence into one equation. Step 3: solve by any method. Step 4: answer the question asked, in words, and check that the answer makes sense (whole numbers, positive ages, and so on).

Typical contexts: prices, ages (remember both people age by the same number of years), two-digit numbers (a number with tens digit $x$ and units digit $y$ is $10x + y$), fares with a fixed charge plus a rate, and fractions.