NCERT Class 9 Maths: Linear Equations in Two Variables - Exercise 4.3

Understanding the Graph of a Linear Equation in Two Variables

A linear equation in two variables, typically written in the form ax + by + c = 0 (where a, b, and c are real numbers, and a and b are not both zero), represents a relationship between two quantities, say x and y. What's truly fascinating is how this relationship translates visually. When you plot all the possible solutions (ordered pairs (x, y) that satisfy the equation) on a Cartesian coordinate plane, they always form a straight line. This is why it's called a 'linear' equation!

Every point on this straight line is a solution to the equation, and conversely, every solution to the equation corresponds to a unique point on that line. Since a straight line extends infinitely in both directions and contains infinitely many points, a linear equation in two variables has infinitely many solutions. To draw the graph of a linear equation, you only need a minimum of two distinct solutions to define the line, but it's good practice to find at least three points. This helps to check for accuracy; if all three points don't lie on the same straight line, it indicates a calculation error.

The graph essentially provides a visual representation of all (x, y) pairs that make the equation true. For example, in the equation x + y = 4, points like (0, 4), (4, 0), (1, 3), (2, 2), (-1, 5) are all solutions and will lie on the same straight line when plotted. Understanding this graphical representation is crucial for solving problems in geometry and for visualizing algebraic relationships.

Key Terms and Concepts

Linear Equation in Two Variables

An equation of the form ax + by + c = 0, where a, b, and c are real numbers, a ≠ 0, b ≠ 0, and x, y are variables. It represents a straight line when graphed.

Solution of a Linear Equation

An ordered pair (x, y) that satisfies the equation, meaning when x and y values are substituted into the equation, the left-hand side equals the right-hand side.

Cartesian Plane

A two-dimensional plane defined by two perpendicular number lines (the x-axis and y-axis) intersecting at the origin (0, 0), used for plotting points.

Graph of a Linear Equation

The collection of all points in the Cartesian plane that represent the solutions of the linear equation. This collection always forms a straight line.

Origin

The point (0, 0) where the x-axis and y-axis intersect on the Cartesian plane. It's often a good point to test for quick solutions.

Steps to Graph a Linear Equation in Two Variables

1. Step 1: Express one variable in terms of the other (Optional but helpful) — If your equation is not already in a form like y = mx + c or x = my + c, rearrange it to make one variable the subject. For example, from 2x + y = 6, you can write y = 6 - 2x. This makes it easier to substitute values and calculate.
2. Step 2: Find at least three solutions (ordered pairs) — Choose at least three convenient values for one variable (e.g., x = 0, 1, 2 or x = -1, 0, 1). Substitute each chosen value into the equation to find the corresponding value of the other variable. This will give you three ordered pairs (x, y) that are solutions to the equation. Using x=0, y=0 (if possible) often yields easy-to-plot points (intercepts).
3. Step 3: Prepare your graph paper — Draw the x-axis (horizontal) and y-axis (vertical) on a Cartesian plane. Label them X, X', Y, Y' and mark the origin (0, 0). Choose an appropriate scale for both axes based on the range of your coordinate values.
4. Step 4: Plot the solutions — Carefully plot each of the three ordered pairs (x, y) you found in Step 2 on your Cartesian plane. Make sure to accurately locate each point based on its x-coordinate and y-coordinate.
5. Step 5: Draw the line — Using a ruler, draw a straight line that passes through all three plotted points. If the points do not align perfectly, recheck your calculations. Extend the line beyond the plotted points and put arrows at both ends to indicate that the line continues infinitely. Finally, write the equation of the line next to it on the graph paper.

Worked Examples: Graphing Linear Equations

• Example 1: Draw the graph of the equation x + y = 5. Step 1: Express y in terms of x: y = 5 - x. Step 2: Find three solutions: If x = 0, then y = 5 - 0 = 5. Point: (0, 5). If x = 5, then y = 5 - 5 = 0. Point: (5, 0). * If x = 2, then y = 5 - 2 = 3. Point: (2, 3). Step 3 & 4: Plot the points (0, 5), (5, 0), and (2, 3) on a Cartesian plane. Step 5: Draw a straight line passing through these points and label it x + y = 5.
• Example 2: Draw the graph of the equation 2x - y = 3. Step 1: Express y in terms of x: y = 2x - 3. Step 2: Find three solutions: If x = 0, then y = 2(0) - 3 = -3. Point: (0, -3). If x = 1, then y = 2(1) - 3 = -1. Point: (1, -1). * If x = 2, then y = 2(2) - 3 = 1. Point: (2, 1). Step 3 & 4: Plot the points (0, -3), (1, -1), and (2, 1) on a Cartesian plane. Step 5: Draw a straight line passing through these points and label it 2x - y = 3.
• Example 3: Draw the graph of the equation y = 3x. Step 1: The equation is already in a convenient form: y = 3x. Step 2: Find three solutions: If x = 0, then y = 3(0) = 0. Point: (0, 0) (This line passes through the origin). If x = 1, then y = 3(1) = 3. Point: (1, 3). * If x = -1, then y = 3(-1) = -3. Point: (-1, -3). Step 3 & 4: Plot the points (0, 0), (1, 3), and (-1, -3) on a Cartesian plane. Step 5: Draw a straight line passing through these points and label it y = 3x.

Common Mistakes and Exam Tips for Graphing

To score full marks in questions involving graphing linear equations, pay close attention to these details:

• Always use at least three points: While two points are sufficient to define a line, using a third point acts as a check. If your three points are not collinear (do not lie on the same straight line), it means there's an error in your calculations or plotting. This simple check can save you from losing marks.
• Accuracy in Plotting: Double-check your coordinate plotting. A slight misplacement of a point can make your "straight line" appear bent. Use a sharp pencil and be precise.
• Extend the Line and Add Arrows: A linear equation has infinite solutions. Therefore, the graph should not stop at your plotted points. Extend the line across the entire graph paper and put arrows at both ends to signify its infinite extent.
• Label Axes and Equation: Always label your x-axis (X, X') and y-axis (Y, Y') clearly, including the origin (0, 0). Most importantly, write the equation of the line next to its graph. Without these labels, your graph is incomplete and harder to interpret.
• Choose Convenient Points: When finding solutions, pick values for x (or y) that result in integer or easily plottable fractional values for the other variable. Avoid large numbers that might go off your graph paper or awkward fractions that are hard to plot accurately.

Practice Questions with Solutions

• Q: Draw the graph of the linear equation x - 2y = 4. A: Step 1: Express x in terms of y (or y in terms of x). Let's use x = 4 + 2y. Step 2: Find three solutions: If y = 0, x = 4 + 2(0) = 4. Point (4, 0). If y = -1, x = 4 + 2(-1) = 2. Point (2, -1). If y = -2, x = 4 + 2(-2) = 0. Point (0, -2). Step 3: Plot the points (4, 0), (2, -1), (0, -2) on a Cartesian plane. Step 4: Draw a straight line passing through these points and label it x - 2y = 4. Final answer: Graph of x - 2y = 4 passing through (4, 0), (2, -1), (0, -2).
• Q: Give the equations of two lines passing through (2, 14). How many more such lines are there, and why? A: Step 1: Identify coordinates: x = 2, y = 14. We need equations where substituting these values holds true. Step 2: Formulate two equations: Equation 1: x + y = k. Substitute x=2, y=14: 2 + 14 = 16. So, x + y = 16 is one equation. Equation 2: 7x - y = k. Substitute x=2, y=14: 7(2) - 14 = 14 - 14 = 0. So, 7x - y = 0 is another equation. Step 3: Determine how many more lines exist. A single point in a plane has infinitely many lines passing through it. Final answer: Two equations could be x + y = 16 and 7x - y = 0. There are infinitely many such lines because a unique line is determined by two distinct points, not just one.
• Q: If the point (3, 4) lies on the graph of the equation 3y = ax + 7, find the value of a. A: Step 1: Substitute the given point (x, y) = (3, 4) into the equation. 3(4) = a(3) + 7 Step 2: Simplify and solve for a. 12 = 3a + 7 12 - 7 = 3a 5 = 3a a = 5/3 Final answer: The value of a is 5/3.
• Q: Draw the graph of the linear equation y = -x. Does it pass through the origin? A: Step 1: Find three solutions: If x = 0, then y = -0 = 0. Point (0, 0). If x = 1, then y = -1. Point (1, -1). If x = -1, then y = -(-1) = 1. Point (-1, 1). Step 2: Plot the points (0, 0), (1, -1), (-1, 1) on a Cartesian plane. Step 3: Draw a straight line passing through these points. Step 4: Observe if (0, 0) is on the line. Since y=0 when x=0, the point (0,0) satisfies the equation. Final answer: The graph of y = -x passes through the origin.

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