Class 9 Maths Chapter 4: Linear Equations In Two Variables Notes

Key Points to Remember

• A linear equation in two variables (x and y) can be written in the standard form: ax + by + c = 0, where a, b, and c are real numbers, and a ≠ 0 or b ≠ 0 (i.e., a and b are not both zero).
• Every linear equation in two variables has infinitely many solutions. Each solution is an ordered pair (x, y) that satisfies the equation.
• The graph of every linear equation in two variables is a straight line.
• Every point on the line is a solution of the linear equation, and every solution of the linear equation is a point on the line.
• An equation of the form y = k (where k is a constant) represents a line parallel to the x-axis.
• An equation of the form x = k (where k is a constant) represents a line parallel to the y-axis.
• The equation x = 0 represents the y-axis, and y = 0 represents the x-axis.
• To draw the graph, find at least two (preferably three to check accuracy) solutions and plot them. Then draw a straight line passing through these points.

Key Definitions

Linear Equation in Two Variables

An algebraic equation of the form ax + by + c = 0, where 'x' and 'y' are variables, and 'a', 'b', 'c' are real numbers such that 'a' and 'b' are not both zero. The highest power of each variable is one.

Solution of a Linear Equation

A pair of values (x, y) that, when substituted into the linear equation, makes the equation true (satisfies the equation).

Standard Form

The conventional way to write a linear equation in two variables, which is ax + by + c = 0, where 'a', 'b', and 'c' are constants and 'x' and 'y' are variables.

Coordinates

A set of values that show an exact position on a coordinate plane. For a point, it's typically an ordered pair (x, y) where 'x' is the abscissa and 'y' is the ordinate.

Abscissa

The x-coordinate of a point in a Cartesian coordinate system, representing its horizontal distance from the y-axis.

Ordinate

The y-coordinate of a point in a Cartesian coordinate system, representing its vertical distance from the x-axis.

Graphical Representation of Linear Equations

One of the most important aspects of linear equations in two variables is their graphical representation. Every linear equation in two variables corresponds to a straight line on a Cartesian coordinate plane. Conversely, every straight line on the plane can be represented by a linear equation in two variables. This powerful connection allows us to visualize algebraic relationships.

To graph a linear equation, you need to find at least two distinct solutions (ordered pairs (x, y)) that satisfy the equation. For example, if you have the equation 2x + y = 6, you can find solutions by choosing values for x and calculating the corresponding y-values:

• If x = 0, then 2(0) + y = 6 ⇒ y = 6. So, (0, 6) is a solution.
• If x = 3, then 2(3) + y = 6 ⇒ 6 + y = 6 ⇒ y = 0. So, (3, 0) is a solution.
• If x = 1, then 2(1) + y = 6 ⇒ 2 + y = 6 ⇒ y = 4. So, (1, 4) is a solution.

Once you have these points (e.g., (0, 6), (3, 0), (1, 4)), you plot them on a graph paper. Since all these points lie on the same straight line, you can draw a line passing through them. This line is the graph of the equation 2x + y = 6. It's crucial to remember that every point on this line is a solution to the equation, and every solution to the equation will lie on this line. This infinite set of points represents the infinite solutions a linear equation in two variables possesses. This visual tool helps in understanding the relationship between the variables more intuitively.

Steps to Graph a Linear Equation

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Worked Examples

• Example 1: Finding Solutions Find two solutions for the equation x + 3y = 9. Solution: 1. Let x = 0: 0 + 3y = 9 ⇒ 3y = 9 ⇒ y = 3. So, (0, 3) is a solution. 2. Let y = 0: x + 3(0) = 9 ⇒ x = 9. So, (9, 0) is a solution. (Other solutions exist, e.g., if x=3, 3+3y=9 => 3y=6 => y=2. So (3,2) is another solution.)
• Example 2: Checking if a point is a solution Check if (2, 5) is a solution to the equation 3x - y = 1. Solution: Substitute x = 2 and y = 5 into the equation: 3(2) - 5 = 6 - 5 = 1. Since LHS = RHS, (2, 5) is a solution to 3x - y = 1.

Section 6

Accuracy in Graphing: When drawing graphs, always use graph paper, a sharp pencil, and a ruler. Mark the scale clearly on both axes. Finding three points instead of two provides a crucial check; if the three points are not collinear, you've made a calculation error, helping you avoid mistakes in exams. Also, label your graph with the equation of the line.

Practice Questions with Solutions

• Q: Write the equation 5x = 2y - 7 in the standard form ax + by + c = 0. A: 5x - 2y + 7 = 0.
• Q: How many solutions does the equation x + y = 10 have? A: Infinitely many solutions.
• Q: What kind of line is represented by the equation y = -4? A: A straight line parallel to the x-axis, 4 units below it.
• Q: Is (1, 1) a solution to the equation 2x + 3y = 5? A: Yes, substituting x=1, y=1 gives 2(1) + 3(1) = 2 + 3 = 5, which satisfies the equation.

Frequently Asked Questions