Linear Equations in Two Variables

1. Introduction

An equation is a statement of equality involving one or more variables. In earlier classes, you studied linear equations in one variable (e.g., $x + 5 = 10$). Now, we will explore equations involving two variables.

2. Definition

A Linear Equation in Two Variables is an equation that can be written in the form:

$ax + by + c = 0$

Where:

$a, b, c$ are real numbers.
$a \neq 0$ and $b \neq 0$ (Both coefficients cannot be zero).
$x$ and $y$ are variables.

Examples:

$2x + 3y = 5$ (Here $a=2, b=3, c=-5$)
$x - 2y - 3 = 0$ (Here $a=1, b=-2, c=-3$)
$y = 3x$ (Can be written as $3x - y + 0 = 0$)

3. Solution of a Linear Equation

A solution to a linear equation in two variables is a pair of values, one for $x$ and one for $y$, which makes the two sides of the equation equal.

Example: For the equation $2x + y = 7$:

If we put $x = 3$ and $y = 1$: LHS = $2(3) + 1 = 6 + 1 = 7$ = RHS. So, $(3, 1)$ is a solution.
If we put $x = 1$ and $y = 5$: LHS = $2(1) + 5 = 7$ = RHS. So, $(1, 5)$ is also a solution.

                    Important Property: A linear equation in two variables has infinitely many
                        solutions.
                

Practice Questions - Free Preview

5 Sample Questions

Question 1

The linear equation $2x - 5y = 7$ has:

A. A unique solution

B. Two solutions

C. Infinitely many solutions

D. No solution

Correct Answer: C
A linear equation in two variables represents a line, and a line contains infinitely many points. Hence, it has infinitely many solutions.

Question 2

The equation $x = 7$ can be written in two variables ($x, y$) as:

A. $1 \cdot x + 0 \cdot y - 7 = 0$

B. $1 \cdot x + 1 \cdot y - 7 = 0$

C. $0 \cdot x + 1 \cdot y - 7 = 0$

D. $1 \cdot x + 0 \cdot y + 7 = 0$

Correct Answer: A
Since there is no $y$ term, its coefficient is 0. Moving 7 to LHS gives $x - 7 = 0$, which is $1 \cdot x + 0 \cdot y - 7 = 0$.

Question 3

If $(2, 0)$ is a solution of the linear equation $2x + 3y = k$, then the value of $k$ is:

A. 4

B. 6

C. 5

D. 2

Correct Answer: A
Substitute $x=2, y=0$ in the equation: $2(2) + 3(0) = k \Rightarrow 4 + 0 = k \Rightarrow k = 4$.

Question 4

Any point on the $x$-axis is of the form:

A. $(x, y)$

B. $(0, y)$

C. $(x, 0)$

D. $(x, x)$

Correct Answer: C
On the $x$-axis, the value of the $y$-coordinate (ordinate) is always 0.

Question 5

Which of the following is NOT a linear equation in two variables?

A. $2x + 3y = 7$

B. $x^2 + 2x = 5$

C. $x = 0$

D. $x - 5y = 2$

Correct Answer: B
Equation B has a term $x^2$, making its degree 2. A linear equation must have degree 1.

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