Pair of Linear Equations in Two Variables · Lesson 1 of 4

Linear Equations in Two Variables

“A masterclass in balancing acts: where two lines either cross paths perfectly, run away together forever, or realize they were the exact same line all along.”

Learning Objectives

• Write a linear equation in the standard form ax + by + c = 0. • Find and verify ordered pairs that satisfy a linear equation in two variables. • Represent a pair of linear equations graphically on a Cartesian coordinate plane. • Relate intersecting, parallel and coincident lines to the number of solutions. • Use coefficient ratios to determine whether a pair has one solution, no solution or infinitely many solutions.

Introduction

Many situations in everyday life involve two quantities that change together. For example, the total cost of buying notebooks depends on both the number of notebooks and their price, while the distance travelled may depend on speed and time. Mathematics allows us to describe relationships like these using equations.

A linear equation in two variables is an algebraic equation involving two unknown quantities, usually represented by x and y. It is called linear because the highest power of each variable is 1.

For example,

2x + 3y = 12

is a linear equation in two variables. Here, both x and y can take different values, but they must work together in such a way that the equation remains true. So, instead of having only one possible answer, a linear equation in two variables can have many pairs of solutions.

Each solution can be written as an ordered pair such as (x, y). When all these solution pairs are plotted on a coordinate plane, they form a straight line. This connection between algebra and graphs makes linear equations especially useful for understanding and representing relationships between two changing quantities.

General form of a linear equation in two variablesLaTeX

a x + b y + c = 0

Symbol	Meaning
x, y	The two variables or unknown quantities
a, b, c	Real-number constants
a, b	The coefficients of x and y

Important condition

The coefficients a and b cannot both be zero. Therefore, a^2+b^2≠0. If both were zero, the equation would not contain either variable.

Real-World Application: The Village Fair Scenario

Imagine visiting a village fair and spending money on two activities: riding a Giant Wheel and playing Hoopla, a game in which a ring is thrown over objects at a stall.

Condition	Information
Relationship	The number of Hoopla games is half the number of Giant Wheel rides.
Cost	Each Giant Wheel ride costs ₹3 and each Hoopla game costs ₹4.
Total spending	The total amount spent is ₹20.

We first assign variables to the two unknown quantities.

First variableLaTeX

x = number of Giant Wheel rides

Second variableLaTeX

y = number of Hoopla games

The number of Hoopla games is half the number of rides. Therefore:

Equation from the first conditionLaTeX

y = \frac{x}{2}

First equation in standard formLaTeX

x - 2 y = 0

The cost of x rides is ₹3 times x, and the cost of y Hoopla games is ₹4 times y. Since the total spending is ₹20:

Equation from the total costLaTeX

3 x + 4 y = 20

What does the solution mean?

The two equations form a pair of linear equations in two variables, also called a system of simultaneous linear equations. Its solution must satisfy both equations at the same time.

Graphical Representation and Geometric Interpretation

Every linear equation in two variables represents a straight line on a Cartesian coordinate plane. Therefore, a pair of linear equations represents two straight lines. These lines can relate to each other in only three ways.

Steps to Solve Graphically

1. Draw the straight line represented by the first equation. 2. Draw the straight line represented by the second equation. 3. Observe the relative positions of the two lines. 4. If the lines intersect, read the coordinates of the point of intersection. A pair of straight lines can have only three possible relationships: intersecting, parallel or coincident.

Worked Example: Solving Graphically

Problem
Solve graphically: 2x+y=7 x-y=2

1.Rewrite each equation in slope-intercept form.
2.Choose convenient values of x to obtain points on each line.
3.Plot the points on graph paper.
4.Draw the two straight lines.
5.Observe their point of intersection.
6.Read the coordinates of the intersection point.

EquationsLaTeX

2 x + y x - y = 7 = 2

x	y = 7 - 2x	Point
0	7	(0,7)
2	3	(2,3)
3	1	(3,1)

First Line— Plot the points (0,7), (2,3) and (3,1) and draw the line representing 2x+y=7.

x	y = x - 2	Point
0	-2	(0,-2)
2	0	(2,0)
3	1	(3,1)

Second Line— Plot the points (0,-2), (2,0) and (3,1) and draw the line representing x-y=2.

Graphical solution — the lines meet at the unique point (3, 1).

Complete Graph— Display both lines on the same Cartesian plane. Highlight the intersection point (3,1).

SolutionLaTeX

x = 3, y = 1

The coordinates of the point of intersection satisfy both equations.

Important Observation

Whenever two straight lines are drawn on the same Cartesian plane, only three geometric relationships are possible: • Intersecting lines — one unique solution. • Parallel lines — no solution. • Coincident lines — infinitely many solutions.

1. Intersecting Lines

Intersecting lines cross each other at exactly one point. The coordinates of this point satisfy both equations, so the pair has exactly one unique solution. Such a system is called consistent.

Exactly One Solution — Consistent Pair

Intersecting Lines— Insert an SVG showing two straight lines intersecting at one point labelled (x,y).

2. Parallel Lines

Parallel lines remain the same distance apart and never meet, even when extended. Since no point lies on both lines, the pair has no solution. Such a system is called inconsistent.

No Solution — Inconsistent Pair

Parallel Lines— Insert an SVG showing two distinct parallel straight lines that never intersect.

3. Coincident Lines

Coincident lines lie exactly on top of each other and represent the same line. Every point on the line satisfies both equations, so the pair has infinitely many solutions. Such a system is dependent and consistent.

Infinitely Many Solutions — Dependent Pair

Coincident Lines— Insert an SVG showing two equations represented by the same overlapping straight line.

The Coefficient Ratio Comparison Method

Drawing a graph is not always necessary. We can predict whether two lines intersect, remain parallel, or coincide by comparing the coefficients of the equations.

First equationLaTeX

a_{1} x + b_{1} y + c_{1} = 0

Second equationLaTeX

a_{2} x + b_{2} y + c_{2} = 0

Compare the three ratios a₁/a₂, b₁/b₂ and c₁/c₂.

Condition	Graphical representation	Number of solutions	System type
a₁/a₂≠b₁/b₂	Intersecting lines	Exactly one	Consistent
a₁/a₂=b₁/b₂=c₁/c₂	Coincident lines	Infinitely many	Dependent and consistent
a₁/a₂=b₁/b₂≠c₁/c₂	Parallel lines	No solution	Inconsistent

Step-by-Step Worked Examples

Example A: Intersecting Lines

Problem
Determine the nature of the lines represented by x-2y=0 and 3x+4y-20=0.

1.Write both equations in the form aᵢx+bᵢy+cᵢ=0.
2.For the first equation: a₁=1, b₁=-2, c₁=0.
3.For the second equation: a₂=3, b₂=4, c₂=-20.
4.Calculate a₁/a₂=1/3 and b₁/b₂=−2/4=-1/2.
5.Since 1/3≠-1/2, we have a₁/a₂≠b₁/b₂.
6.Therefore, the lines intersect and the pair has one unique solution.

Coefficient-ratio checkLaTeX

\frac{a _{1}}{a _{2}} = \frac{1}{3} \neq = - \frac{1}{2} = \frac{b _{1}}{b _{2}}

The first two ratios are unequal, so the lines intersect.

Example B: Coincident Lines

Problem
Determine the nature of the lines represented by 2x+3y-9=0 and 4x+6y-18=0.

1.For the first equation: a₁=2, b₁=3, c₁=-9.
2.For the second equation: a₂=4, b₂=6, c₂=-18.
3.Calculate a₁/a₂=2/4=1/2.
4.Calculate b₁/b₂=3/6=1/2.
5.Calculate c₁/c₂=−9/−18=1/2.
6.All three ratios are equal, so the lines are coincident and the pair has infinitely many solutions.

Coefficient-ratio checkLaTeX

\frac{a _{1}}{a _{2}} = \frac{b _{1}}{b _{2}} = \frac{c _{1}}{c _{2}} = \frac{1}{2}

All three ratios are equal, so the lines are coincident.

Example C: Parallel Lines

Problem
Determine the nature of the lines represented by x+2y-4=0 and 2x+4y-12=0.

1.For the first equation: a₁=1, b₁=2, c₁=-4.
2.For the second equation: a₂=2, b₂=4, c₂=-12.
3.Calculate a₁/a₂=1/2 and b₁/b₂=2/4=1/2.
4.Calculate c₁/c₂=−4/−12=1/3.
5.Thus, a₁/a₂=b₁/b₂≠c₁/c₂.
6.Therefore, the lines are parallel and the pair has no solution.

Coefficient-ratio checkLaTeX

\frac{a _{1}}{a _{2}} = \frac{b _{1}}{b _{2}} = \frac{1}{2} \neq = \frac{1}{3} = \frac{c _{1}}{c _{2}}

The first two ratios are equal, but the constant-term ratio is different.

Quick checking order

First compare a₁/a₂ and b₁/b₂. If they are unequal, the lines intersect and you can stop. Compare c₁/c₂ only when the first two ratios are equal.

Quick check

Which is the correct general form of a linear equation in two variables?

Quick check

Which ordered pair satisfies both 2x + y = 7 and x − y = 2?

Quick check

Which coefficient-ratio condition shows that two lines intersect at exactly one point?

Quick check

What is the relationship between the lines x + 2y − 4 = 0 and 2x + 4y − 12 = 0?

Quick check

The equations 2x + 3y − 9 = 0 and 4x + 6y − 18 = 0 represent which type of lines?

Practice Problems

Write 3x + 2y = 12 in the form ax + by + c = 0. Identify a, b and c, and find any two ordered pairs that satisfy the equation.
Solve the pair x + y = 5 and x − y = 1 graphically. State the point at which the two lines intersect.
Determine the nature of the lines represented by 2x − 3y + 5 = 0 and 4x − 6y + 10 = 0. Also state the number of solutions.
Determine the nature of the lines represented by 3x + 2y − 7 = 0 and 6x + 4y + 5 = 0. Also state whether the system is consistent or inconsistent.
Find the values of k for which the equations 2x + 3y − 5 = 0 and 4x + 6y − k = 0 have no solution. What happens when k = 10?

Key Takeaways

• A linear equation in two variables has the general form ax + by + c = 0, where a and b cannot both be zero. • Every solution is an ordered pair (x, y), and the graph of all its solutions is a straight line. • Two intersecting lines have exactly one solution and form a consistent system. • Parallel lines have no solution, while coincident lines have infinitely many solutions. • Coefficient ratios can identify the relationship between two lines without drawing their graphs.

Start of chapter

Next · Lesson 2

Algebraic Method 1: The Substitution Method

Introduction

Real-World Application: The Village Fair Scenario

Graphical Representation and Geometric Interpretation

1. Intersecting Lines

2. Parallel Lines

3. Coincident Lines

The Coefficient Ratio Comparison Method

Step-by-Step Worked Examples

Quiz

Practice Problems

Key Takeaways