CBSE Class 9 Maths Notes: Linear Equations in Two Variables

Review of Linear Equations in One Variable

Let’s revisit what you already know! A linear equation in one variable involves only one unknown (usually represented by ‘x’) and the highest power of the variable is 1. These equations have a single solution, a value of $x$ that satisfies the equation.

Definition: A linear equation in one variable is an equation that can be written in the form $ax + b = 0$, where $a$ and $b$ are real numbers, and $a \ne 0$.

Example: $2x + 5 = 0$. Solving this gives $x = -\frac{5}{2}$, which is a single point on the number line.

Core Principle: The solution of a linear equation in one variable is a unique value of the variable.

Form of a Linear Equation in Two Variables: $ax + by + c = 0$

Linear equations in two variables involve two unknowns, typically $x$ and $y$. They take the general form $ax + by + c = 0$, where $a$, $b$, and $c$ are real numbers, and crucially, $a$ and $b$ are *not* both zero simultaneously. This form helps us understand the fundamental structure of these equations.

Definition: The standard form of a linear equation in two variables is $ax + by + c = 0$, where $a, b,$ and $c$ are real numbers, and $a$ and $b$ are not both zero.

Examples:

• $2x + 3y – 6 = 0$ (Here, $a=2$, $b=3$, and $c=-6$)
• $y = 2x + 1$ (This can be rewritten as $-2x + y – 1 = 0$)

Solutions as Ordered Pairs

Unlike linear equations in one variable, linear equations in two variables have infinitely many solutions. Each solution is an ordered pair of numbers $(x, y)$ that makes the equation true. These pairs represent points on a coordinate plane.

Core Principle: A solution to a linear equation in two variables is an ordered pair $(x, y)$ that, when substituted into the equation, makes the equation true.

Example: Consider the equation $x + y = 5$. Some solutions are $(2, 3)$, $(0, 5)$, and $(5, 0)$. Substituting $x=2$ and $y=3$ gives $2 + 3 = 5$, which is true.

Graphing on the Cartesian Plane

Visualizing linear equations is key! Each linear equation in two variables represents a straight line on the Cartesian plane. The x-coordinate and y-coordinate of any point on this line constitute a solution of the equation.

Core Principle: The graph of a linear equation in two variables is a straight line. Every point on the line represents a solution to the equation.

Steps for Graphing:

1. Rewrite the equation in slope-intercept form ($y = mx + b$) if necessary.
2. Find at least two solutions (ordered pairs) to the equation. A good starting point is often finding the x-intercept and y-intercept.
3. Plot the points on the Cartesian plane.
4. Draw a straight line through the plotted points.

Example: To graph $2x + y = 4$, first rewrite it as $y = -2x + 4$. Find two points: When $x=0$, $y=4$, and when $y=0$, $x=2$. Plot $(0, 4)$ and $(2, 0)$ and draw a line through them.

Understanding Infinitely Many Solutions

The straight line representing a linear equation in two variables extends infinitely in both directions. This means there are an infinite number of points (and therefore solutions) along the line.

Core Principle: A linear equation in two variables has infinitely many solutions because there are infinitely many points on the line that represents it.

Connection to the Graph: Each point on the line corresponds to a unique ordered pair $(x, y)$ that satisfies the equation. Since the line continues indefinitely, the number of solutions is unbounded.

Exploring Collinearity Using Plotted Points

Collinear points are points that lie on the same straight line. We can use the concept of linear equations to determine if a set of points is collinear. If the coordinates of the points satisfy the equation of a line, then the points are collinear.

Core Principle: If three or more points satisfy the same linear equation, then they are collinear.

Method:

1. Find the equation of the line passing through two of the given points.
2. Substitute the coordinates of the remaining points into the equation.
3. If all the remaining points satisfy the equation, then all points are collinear.

Example: Determine if points $A(1, 1)$, $B(2, 3)$, and $C(3, 5)$ are collinear.
Find the equation using $A$ and $B$, $y = 2x – 1$. Substitute $C(3,5)$ into the equation. $5 = 2(3) – 1$, simplifies to $5=5$. Therefore, the points are collinear.