Chapter 3: Pair of Linear Equations in Two Variables

A single equation like $2x + 3y = 12$ has infinitely many solutions , pick any $x$, the equation forces a $y$. To pin down a unique pair of values, we need a second equation. This chapter is the study of two such equations considered together and the methods to find the values of $x$ and $y$ that satisfy both.

The geometry is clean. Each linear equation is a straight line in the $xy$-plane. A pair of equations is a pair of lines. Two lines in a plane either intersect in one point, are parallel and never meet, or are the same line (overlap). These three cases give us three solution behaviours: a unique solution, no solution, or infinitely many solutions. The chapter is essentially a fluent translation between this geometric picture and three algebraic techniques: graphing, substitution, and elimination.

For boards, expect a mix: $2$- to $3$-mark direct solving of a pair, $3$-mark consistency / no-solution analysis, and a juicy $4$- to $5$-mark word problem on age, speed, time and distance, work, fractions, or numbers with two unknowns. Many real-life problems , mixture composition, business break-even, supply/demand , boil down to a pair of linear equations.

We will favour clarity over cleverness. Substitution is the safest tool when one variable already has a coefficient of $\pm 1$. Elimination is the cleanest tool when the coefficients are not too friendly. The graphical method, while not always exam-efficient, gives the deepest intuition.

What's inside

• Graphical method , plotting two lines and reading the solution.
• Consistency: a coefficient-only test , when the system has a unique, no, or infinite solutions.
• Substitution method , algebraic muscle for clean elimination of a variable.
• Elimination method , adding or subtracting multiples to cancel.
• Word problems , translating English into a pair of equations and back.

Key results / Formula card

For the system $a_1 x + b_1 y + c_1 = 0,$ $a_2 x + b_2 y + c_2 = 0,$

The first row corresponds to a consistent independent system; the second to an inconsistent system; the third to a consistent dependent system.