Chapter 3 · Introduction

Chapter 3 — Pair of Linear Equations in Two Variables

When two unknowns are tied together by two conditions, each condition is a straight line and the answer is where the lines meet. Learn to find that point by drawing, by substitution and by elimination.

You open two cab apps for a ride across the city. One charges a higher booking fee but less per kilometre; the other has a small booking fee but charges more per kilometre. Which is cheaper? The honest answer is "it depends on how far you go". Each fare rule is a straight-line graph of cost against distance, and the two lines cross at one special distance. For shorter rides one app wins, for longer rides the other does. Finding that crossing point means solving a pair of linear equations.

Pairs of equations turn up whenever two unknown quantities are linked by two facts: the prices of two items from two bills, the ages of a parent and child, the speeds of two cyclists, the digits of a number. In this chapter you will learn three ways to solve them. The graphical method shows what is going on: two lines that cross, run parallel, or lie on top of each other. The substitution and elimination methods give exact answers quickly, even when the solution is an awkward fraction that is hard to read off a graph. Finally you will turn word problems into equations and solve them.

Each linear equation is a straight line→A solution is a point on both lines→The ratios $\tfrac{a_1}{a_2}, \tfrac{b_1}{b_2}, \tfrac{c_1}{c_2}$ predict one, none or infinitely many→Substitution and elimination find it exactly

🎯 By the end of this chapter you can…

write a pair of linear equations in the general form $a_1x+b_1y+c_1=0$, $a_2x+b_2y+c_2=0$ and check whether a given point is a solution;
solve a pair graphically from tables of points, and recognise intersecting, parallel and coincident lines;
use the ratios $\dfrac{a_1}{a_2}$, $\dfrac{b_1}{b_2}$, $\dfrac{c_1}{c_2}$ to decide, without drawing, whether a pair has one solution, no solution or infinitely many, and find an unknown coefficient $k$ from such a condition;
solve a pair exactly by the substitution method and by the elimination method, and choose the quicker one;
turn word problems about prices, ages, money, digits, fractions and speeds into a pair of equations, solve them, and check the answer in the original words.

Sections in this chapter

3.1

Pairs of Linear Equations and the Graphical Method

Solutions as meeting points of two lines, solving from tables of points, and the ratio test for one, none or infinitely many solutions.

$\tfrac{a_1}{a_2}\ne\tfrac{b_1}{b_2}\Rightarrow$ one solution

Open section →3.2

The Substitution Method

Make one variable the subject of one equation, put it into the other, and solve a single equation in one unknown.

$y=7-x\ \Rightarrow\ x-(7-x)=1$

Open section →3.3

The Elimination Method and Word Problems

Multiply to match a pair of coefficients, add or subtract to remove one variable, and model problems on ages, money, digits, fractions and speed.

$(x+y)+(x-y)=7+1\Rightarrow 2x=8$

Open section →

✅ Before you start

From Class 9 Linear Equations in Two Variables you need: what it means for $(x, y)$ to be a solution of an equation such as $2x+3y=12$; how to find solutions by choosing $x$ and working out $y$; and the fact that the graph of a linear equation in two variables is a straight line. From Class 9 Coordinate Geometry you need to plot points such as $(3,-2)$ accurately on graph paper. You should also be confident solving a linear equation in one variable, like $5x-3=2x+9$ (Class 8), and simplifying expressions with brackets and negative signs. The zeroes of a linear polynomial from Section 2.1 — where a line crosses the $x$-axis — will help when you draw tables of points.

← 2.2 Relationship between Zeroes and Coefficients 3.1 Pairs of Linear Equations and the Graphical Method →