Linear Equations in Two Variables
CBSE Class 9 · Mathematics · Notes, formulas and practice questions
Revise linear equations in two variables for CBSE Class 9: solutions as ordered pairs, infinitely many solutions, and drawing their straight-line graphs, including lines parallel to the axes.
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A linear equation in two variables has the form ax + by + c = 0. It connects x and y but does not decide both at once, so a solution is an ordered pair (x, y) that makes the equality true. For example, x + y = 7 is true for (0,7), (1,6) and also for (0.5, 6.5). Choosing any value for one variable and then calculating the other gives one pair; because the choice is free, there are infinitely many solutions.
The visual idea carries the chapter: all solutions of one linear equation lie on a single straight line. To draw the graph, find two convenient solutions, plot them as points, and join them with a straight edge; a third point can verify the line. To see where the line meets the x-axis, set y = 0; to see where it meets the y-axis, set x = 0. Every point on that line is a solution, and every solution is a point on the line.
Some linear equations look as though they use only one variable, such as x = 6 or y = -1/4. They are still equations in two variables because the missing variable carries coefficient 0: x = 6 can be written x + 0y = 6, and y = -1/4 as 0x + y = -1/4. Their graphs are straight lines parallel to an axis: x = k is vertical, and y = k is horizontal.
This chapter deals with exactly one equation, not a pair. When a suggested pair is given, substitute it into the single equation to check whether it is a solution. Do not set up two equations or look for intersections of two lines; that belongs to Pair of Linear Equations in Class 10. Practice rewriting equations in general form and recognising the kind of line from its equation.
Key terms
- Linear equation in two variables
- An equation that can be written as ax + by + c = 0, where x and y are variables of exponent 1 and a, b, c are real numbers with a and b not both zero. Its graph is always a straight line.
- Solution as an ordered pair
- A solution is an ordered pair (x, y) that satisfies the equation, meaning the left side equals the right side after substitution. The first number is x and the second is y.
- Infinitely many solutions
- A single linear equation in two variables has infinitely many ordered-pair solutions. You can choose a value for one variable, substitute it, and solve for the other, which remains possible for every real number you choose.
- General form ax + by + c = 0
- The standard way to write such an equation with all terms on one side. For example, y = 3x − 2 becomes 3x − y − 2 = 0 after rearranging. Multiplying both sides by a non-zero constant gives the same line.
- Graph of a linear equation
- The set of all points whose coordinates are solutions. Because the equation is linear, these points lie on one straight line, so plotting two solutions and joining them is enough to draw the full graph.
- Line x = k
- The graph of x = k is a vertical line parallel to the y-axis, passing through (k, 0). If k = 0, it is exactly the y-axis. All points on it have the same x-coordinate k and any y-coordinate.
- Line y = k
- The graph of y = k is a horizontal line parallel to the x-axis, passing through (0, k). If k = 0, it is exactly the x-axis. All points on it have the same y-coordinate k and any x-coordinate.
- Intercepts of a line
- The x-intercept is where the graph crosses the x-axis, so its y-coordinate is 0; the y-intercept is where it crosses the y-axis, so its x-coordinate is 0. These points are useful for drawing the graph quickly.
Formula sheet
| What | Formula | Notes |
|---|---|---|
| General linear equation | ax + by + c = 0 | a, b and c are real numbers; a and b are not both zero. This form is the standard one used for checking whether an ordered pair is a solution. |
| Express y in terms of x | y = (−ax − c)/b | Use this when b ≠ 0. Put any chosen x-value into the right side and you get the matching y-value of a solution. If b = 0, solve for x instead. |
| Lines parallel to the coordinate axes | x = k, y = k | k is any real number. The equation x = k gives a vertical line parallel to the y-axis; y = k gives a horizontal line parallel to the x-axis. |
| Intercepts of ax + by + c = 0 | x-intercept = −c/a, y-intercept = −c/b | Works when a ≠ 0 and b ≠ 0. Set y = 0 to find the x-intercept and x = 0 to find the y-intercept. The line crosses the axes at (−c/a, 0) and (0, −c/b). |
Practice questions with answers
1. Which of (1,2), (−1,4) and (0,4) are solutions of 2x + y − 4 = 0?
Check each pair by substitution. For (1,2): 2×1 + 2 − 4 = 0, so it is a solution. For (−1,4): 2×(−1) + 4 − 4 = −2, not a solution. For (0,4): 2×0 + 4 − 4 = 0, so it is a solution. Thus (1,2) and (0,4) satisfy the equation.
2. Write 2y − 3x = 5 in the general form ax + by + c = 0 and find two solutions.
Bring all terms to one side: −3x + 2y − 5 = 0. Multiplying by −1 gives 3x − 2y + 5 = 0. Choose x = 1: 3 − 2y + 5 = 0, giving y = 4, so (1,4) is a solution. Choose x = 3: 9 − 2y + 5 = 0, giving y = 7, so (3,7) is another solution.
3. For what value of k is x = 2, y = 5 a solution of y = kx + 3?
Substitute x = 2 and y = 5 into y = kx + 3. This gives 5 = 2k + 3, so 2k = 2 and k = 1. The equation is y = x + 3.
4. What is the graph of x = −2, and why is it a linear equation in two variables?
The graph of x = −2 is a vertical line parallel to the y-axis through (−2, 0). It can be written as x + 0y + 2 = 0, which is of the form ax + by + c = 0 with a = 1 and b = 0. Every point (−2, y) is a solution.
5. Write the equation of a line parallel to the x-axis and 3 units above it.
A line parallel to the x-axis has all its points with the same y-coordinate. Three units above the x-axis means y = 3, so the equation is y = 3. Its graph is horizontal.
6. On the line 2x − 3y = 9, find the y-coordinate of the point whose x-coordinate is 6.
Put x = 6 into the equation: 2×6 − 3y = 9, so 12 − 3y = 9. This gives −3y = −3, so y = 1. The required point is (6, 1).
7. Describe how to draw the graph of x + y = 5.
The line cuts the x-axis when y = 0, giving x = 5, and the y-axis when x = 0, giving y = 5. Plot (5,0) and (0,5), then join them with a straight line. The graph is the line through these two points.
8. State whether the graph of y = 0 is the x-axis. Give a reason.
Yes. The equation y = 0 is satisfied by (x, 0) for every real x. Since the x-axis consists exactly of points whose y-coordinate is 0, the graph of y = 0 is the x-axis.
9. If the point (p, 2p) lies on the line x + y = 6, find p.
Substitute x = p and y = 2p into x + y = 6. Then p + 2p = 6, so 3p = 6 and p = 2. The point is (2,4).
10. Why is x = 5 considered a linear equation in two variables even though y does not appear?
It can be written as x + 0y − 5 = 0, matching ax + by + c = 0 with a = 1, b = 0 and c = −5. The coefficient of y is zero but the equation still has infinitely many solutions of the form (5, y).
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