Linear Equations in Two Variables: Class 9 Study Guide
An equation such as 2x + 3y = 12 has many pairs of numbers that satisfy it, and they all lie on a straight line.
A linear equation in two variables has the form ax + by + c = 0, where a and b are not both zero. A solution is a pair (x, y) that makes the equation true. The equation 2x + 3y = 12 has the solutions (0, 4), (6, 0) and (3, 2), and infinitely many more.
If you plot all these solutions on the Cartesian plane, they lie on a straight line. The line is the graph of the equation, and every point on it is a solution.
Making a table of solutions
For 2x + 3y = 12, choose a value of x and solve for y. When x = 0, 3y = 12, so y = 4. When x = 3, 3y = 6, so y = 2. When x = 6, 3y = 0, so y = 0.
| x | 0 | 3 | 6 |
|---|---|---|---|
| y | 4 | 2 | 0 |
Drawing the graph
Plot at least three points from the table. Two points are enough to fix a line, and the third is a check. Join them with a ruler and extend the line with arrows at both ends.
Choose easy values of x.
Solve the equation.
Check all lie in a line.
Label it with its equation.
Special lines
The equation x = 3 is a vertical line through (3, 0), and y = −2 is a horizontal line through (0, −2). An equation such as y = 2x passes through the origin. If a notebook costs twice as much as a pen, then with pen price x and notebook price y, y = 2x describes the possible prices.
Worked examples and common mistakes
Example: Is (2, 1) a solution of x + 2y = 4? 2 + 2 = 4, so yes. Example: Write x = 5 as an equation in two variables: x + 0y − 5 = 0. Common mistakes: thinking that a single pair is the only solution, drawing a line with only two points and no check, and mixing up x and y when plotting.
Check your understanding: model answers
1. 12 + y = 14, so y = 2. 2. 2 + 2 = 4, which is not 5, so it is not a solution. 3. For example (0, 0), (1, 1) and (2, 2).
✨ Key Takeaways
- A linear equation in two variables has infinitely many solutions.
- Its graph is a straight line.
- Two points fix a line, and a third checks it.
- x = a is vertical and y = b is horizontal.
💬 Questions for Discussion & Reflection
- Find y when x = 4 in 3x + y = 14.
- Is (1, 2) a solution of 2x + y = 5?
- Write the graph of y = x as a table of three points.
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