Linear Equation in Two Variables: A Complete Guide for Class 9

What is a Linear Equation in Two Variables?

An equation that can be written in the form ax + by + c = 0 is called a linear equation in two variables. Let's break this down:

• x and y are the two variables. Their values can change.
• a, b, and c are real numbers, called coefficients and the constant term, respectively. For example, in the equation 3x + 4y - 7 = 0, we have a = 3, b = 4, and c = -7.
• The most important condition is that a and b are not both zero. If both were zero, we'd be left with c = 0, which doesn't have any variables! The term 'linear' means that the highest power of the variables x and y is 1. You won't see terms like x² or y³ in a linear equation. Every solution (x, y) to this equation is a point that lies on the line representing the equation.

How to Find Solutions for a Linear Equation

1. Understand the Core Idea — A linear equation in two variables has infinitely many solutions. A 'solution' is a pair of values, one for x and one for y, that makes the equation true. We write it as an ordered pair (x, y).
2. Step 1: Choose a Value for One Variable — Pick any value for either x or y. The easiest choice is often x = 0 or y = 0.
3. Step 2: Substitute the Value — Plug the value you chose into the equation. For example, let's find a solution for 2x + y = 7. Let's choose x = 1.
4. Step 3: Solve for the Other Variable — After substituting, you'll have a simple linear equation in one variable. Solve it. Using our example: 2(1) + y = 7 becomes 2 + y = 7. Subtracting 2 from both sides gives y = 5.
5. Step 4: Write the Solution — Write the values as an ordered pair (x, y). So, for our example, one solution is (1, 5). You can repeat this process with different starting values to find more solutions, like (0, 7) or (3, 1).

Worked Example: Graphing a Linear Equation

• Problem: Draw the graph of the equation x + 2y = 6. Step 1: Find at least three solutions. Finding three points is a good way to check your work. If all three don't lie on a straight line, there's a calculation error. Let x = 0: 0 + 2y = 6 2y = 6 y = 3 Solution 1: (0, 3) Let y = 0: x + 2(0) = 6 x = 6 Solution 2: (6, 0) * Let x = 2: 2 + 2y = 6 2y = 4 y = 2 Solution 3: (2, 2) Step 2: Create a solution table. | x | 0 | 6 | 2 | |---|---|---|---| | y | 3 | 0 | 2 | Step 3: Plot the points on a Cartesian plane. Plot the points (0, 3), (6, 0), and (2, 2) on the graph paper. Step 4: Draw a straight line through the points. Use a ruler to connect the points with a straight line. Extend the line in both directions and add arrows at the ends to show it continues infinitely. This line represents all the infinite solutions of the equation x + 2y = 6.

Common Mistakes to Avoid

Pay close attention to these common errors that students make in exams:

1. Mixing up Coordinates: Always remember that an ordered pair is (x, y). The x-coordinate comes first (horizontal axis), and the y-coordinate comes second (vertical axis). A common mistake is plotting (2, 3) at the position of (3, 2).

1. Assuming Only One Solution: Unlike equations with one variable, a linear equation in two variables has infinite solutions. Don't stop after finding just one unless the question specifically asks for it.

1. Calculation Errors: When substituting a value to find the other, be very careful with positive and negative signs. A small error here will lead to a point that doesn't lie on the line.

Pro Tip for Graphing: Always find three points. If all three points lie on a single straight line, your calculations are most likely correct. If they don't, you need to go back and check your work!

Practice Questions with Solutions

• Q: The cost of a notebook is ₹5 less than twice the cost of a pen. Write a linear equation in two variables to represent this statement. (Take the cost of a notebook to be ₹x and the cost of a pen to be ₹y). A: Step 1: Let the cost of a notebook be ₹x and the cost of a pen be ₹y. Step 2: According to the statement, "twice the cost of a pen" is 2y. "₹5 less than twice the cost of a pen" is 2y - 5. Step 3: The cost of the notebook (x) is equal to this expression. So, x = 2y - 5. Step 4: To write it in the standard form ax + by + c = 0, we rearrange the terms: x - 2y + 5 = 0. Final answer: The required linear equation is x - 2y + 5 = 0.
• Q: Find four different solutions for the equation x + 3y = 7. A: Step 1: Choose a value for x or y and solve for the other. Let's start with x = 1. 1 + 3y = 7 => 3y = 6 => y = 2. First solution is (1, 2). Step 2: Let y = 0. x + 3(0) = 7 => x = 7. Second solution is (7, 0). Step 3: Let x = 4. 4 + 3y = 7 => 3y = 3 => y = 1. Third solution is (4, 1). Step 4: Let y = 3. x + 3(3) = 7 => x + 9 = 7 => x = -2. Fourth solution is (-2, 3). Final answer: Four solutions are (1, 2), (7, 0), (4, 1), and (-2, 3).
• Q: Check which of the following are solutions of the equation 2x - y = 6: (i) (3, 0) (ii) (0, -6) (iii) (4, 2) A: Step 1: To check a solution, substitute the x and y values from the ordered pair into the equation. If the Left Hand Side (LHS) equals the Right Hand Side (RHS), it is a solution. Step 2: Check (i) (3, 0). LHS = 2(3) - 0 = 6 - 0 = 6. Since LHS = RHS (6), it is a solution. Step 3: Check (ii) (0, -6). LHS = 2(0) - (-6) = 0 + 6 = 6. Since LHS = RHS (6), it is a solution. Step 4: Check (iii) (4, 2). LHS = 2(4) - 2 = 8 - 2 = 6. Since LHS = RHS (6), it is a solution. Final answer: All three points (3, 0), (0, -6), and (4, 2) are solutions to the equation.
• Q: Write the equation y/3 - 5 = 2x in the form ax + by + c = 0 and indicate the values of a, b, and c. A: Step 1: Start with the given equation: y/3 - 5 = 2x. Step 2: To eliminate the fraction, multiply the entire equation by 3. This gives: y - 15 = 6x. Step 3: Rearrange the terms to match the standard form ax + by + c = 0. Move all terms to one side. Let's move them to the right side to keep x positive: 0 = 6x - y + 15. Step 4: Rewrite it in standard form: 6x - y + 15 = 0. Now, compare this with ax + by + c = 0. a = 6, b = -1, c = 15. Final answer: The equation is 6x - y + 15 = 0, where a = 6, b = -1, and c = 15.

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