Pair of Linear Equations in Two Variables — chapter notes

An equation of the form $ax + by = c$, where $a$, $b$, $c$ are real numbers and $a$, $b$ are not both zero, is a linear equation in two variables. A solution is an ordered pair $(x, y)$ that satisfies it.

One linear equation has infinitely many solutions: choose any $x$ and solve for $y$. All of them lie on one straight line, which is the graph of the equation. Two points determine the line; a third is a check. The easiest points are the intercepts: put $x = 0$, then $y = 0$.

A pair of linear equations $a_1x + b_1y = c_1$, $a_2x + b_2y = c_2$ is solved by finding the point(s) common to both lines.

Method: make a table of at least two (preferably three) points for each equation, plot them on the same axes, draw both lines, read the point of intersection, and check it in both equations.

Example: $x + y = 5$ and $2x - y = 4$ meet at $(3, 2)$.

The lines may intersect (one solution), be parallel (no solution) or coincide (infinitely many solutions).

For $a_1x + b_1y = c_1$ and $a_2x + b_2y = c_2$:

• $\dfrac{a_1}{a_2} \ne \dfrac{b_1}{b_2}$: intersecting lines, unique solution (consistent).
• $\dfrac{a_1}{a_2} = \dfrac{b_1}{b_2} \ne \dfrac{c_1}{c_2}$: parallel lines, no solution (inconsistent).
• $\dfrac{a_1}{a_2} = \dfrac{b_1}{b_2} = \dfrac{c_1}{c_2}$: coincident lines, infinitely many solutions (consistent, dependent).

To find an unknown $k$, write the condition for the required case and solve; for "no solution" also check that $\dfrac{c_1}{c_2}$ is different. Example: $2x + 3y = 5$ and $4x + ky = 12$ have no solution for $k = 6$, since $\dfrac{5}{12} \ne \dfrac12$.

1. From one equation, express one variable in terms of the other (choose a variable with coefficient $1$ or $-1$ if possible).
2. Substitute into the other equation, using brackets.
3. Solve the one-variable equation.
4. Back-substitute to find the other variable, and check in both equations.

Example: $3x + y = 11$, $2x - 3y = 0$: $y = 11 - 3x$; $2x - 3(11 - 3x) = 0 \Rightarrow 11x = 33 \Rightarrow x = 3$, $y = 2$.

With fractions, first multiply each equation by the LCM of its denominators.

1. Multiply one or both equations by non-zero numbers so that the coefficients of one variable become equal or opposite.
2. Subtract (equal coefficients) or add (opposite coefficients) to eliminate that variable.
3. Solve, back-substitute, check.

Example: $4x + 3y = 29$, $3x - 2y = 9$: multiply by $2$ and $3$, add: $17x = 85$, $x = 5$, $y = 3$.

If both variables vanish: a false statement (e.g. $0 = 4$) means no solution; a true statement ($0 = 0$) means infinitely many solutions.

Shortcut for swapped coefficients ($37x + 43y = 123$, $43x + 37y = 117$): add and subtract to get $x + y = 3$ and $x - y = -1$.

Not in the 2026–27 syllabus: the cross-multiplication method, and equations reducible to a pair of linear equations.

Steps: (1) define the two unknowns with units; (2) write one equation for each condition; (3) solve by substitution or elimination; (4) answer in words and check in the original statement.

Common translations:

• Money: quantity equation ($x + y =$ total number) and value equation ($10x + 20y =$ total amount).
• Ages: "$n$ years ago" subtracts $n$ from both ages; "after $n$ years" adds $n$ to both.
• Digits: a two-digit number with tens digit $t$ and units digit $u$ is $10t + u$; reversed it is $10u + t$.
• Fractions: write $\dfrac nd$ and cross-multiply each condition.
• Mixtures and investments: amount equation and value (or interest) equation.