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Class 9 — Mathematics (NCERT)

Linear Equations in Two Variables

Class 9

  • ✓Define a linear equation in two variables and identify its standard form.
  • ✓Understand what constitutes a solution to a linear equation in two variables.
  • ✓Find multiple solutions for a given linear equation in two variables.
  • ✓Plot the graph of a linear equation in two variables on the Cartesian plane.
  • ✓Recognise that the graph of a linear equation in two variables is always a straight line.

Key concepts

Linear Equation in Two Variables

An equation which can be written in the form ax + by + c = 0, where a, b and c are real numbers, and a and b are not both zero, is called a linear equation in two variables. Here, 'x' and 'y' are the two variables. The highest power of each variable in such an equation is 1.

ax + by + c = 0
Solution of a Linear Equation in Two Variables

A pair of values, one for 'x' and one for 'y', which satisfies the given linear equation is called a solution of the equation. Since there are two variables, the solution is always an ordered pair (x, y). A linear equation in two variables has infinitely many solutions.

Graph of a Linear Equation in Two Variables

The collection of all points (x, y) in the Cartesian plane that represent the solutions of a linear equation in two variables forms a straight line. This straight line is called the graph of the linear equation. To draw the graph, we need to find at least two solutions, plot these points on the Cartesian plane, and then draw a straight line passing through them. It is advisable to find three solutions to ensure accuracy; if the three points are not collinear, there is an error in calculation.

Key facts to remember

  • 1A linear equation in two variables can be written in the standard form ax + by + c = 0, where a, b, c are real numbers and a, b are not both zero.
  • 2Every linear equation in two variables has infinitely many solutions.
  • 3Each solution of a linear equation in two variables is an ordered pair (x, y).
  • 4The graph of every linear equation in two variables is a straight line.
  • 5Every point on the graph of a linear equation is a solution of the equation.
  • 6Every solution of a linear equation is a point on its graph.
  • 7Equations of the form x = k represent a vertical line parallel to the Y-axis.
  • 8Equations of the form y = k represent a horizontal line parallel to the X-axis.

Worked examples

Example 1

Find four different solutions for the equation 2x + y = 7.

IGiven equation is 2x + y = 7.
IIWe can express y in terms of x: y = 7 - 2x.
IIILet's choose some values for x and find the corresponding values for y:
IVCase 1: Let x = 0.
VSubstitute x = 0 into y = 7 - 2x:
VIy = 7 - 2(0)
VIIy = 7 - 0
VIIIy = 7. So, (0, 7) is a solution.
9Case 2: Let x = 1.
10Substitute x = 1 into y = 7 - 2x:
11y = 7 - 2(1)
12y = 7 - 2
13y = 5. So, (1, 5) is a solution.
14Case 3: Let x = 2.
15Substitute x = 2 into y = 7 - 2x:
16y = 7 - 2(2)
17y = 7 - 4
18y = 3. So, (2, 3) is a solution.
19Case 4: Let x = -1.
20Substitute x = -1 into y = 7 - 2x:
21y = 7 - 2(-1)
22y = 7 + 2
23y = 9. So, (-1, 9) is a solution.

Answer

Four different solutions for the equation 2x + y = 7 are (0, 7), (1, 5), (2, 3), and (-1, 9).

There are infinitely many solutions; these are just four examples.

Example 2

Draw the graph of the equation x - y = 2.

IGiven equation is x - y = 2.
IIWe can express y in terms of x: y = x - 2.
IIILet's find at least three solutions for the equation:
IVCase 1: Let x = 0.
Vy = 0 - 2 = -2. So, (0, -2) is a solution.
VICase 2: Let x = 2.
VIIy = 2 - 2 = 0. So, (2, 0) is a solution.
VIIICase 3: Let x = 4.
9y = 4 - 2 = 2. So, (4, 2) is a solution.
10Now, we plot these points (0, -2), (2, 0), and (4, 2) on the Cartesian plane.
11Draw a straight line passing through these three points.
12Label the line with its equation, x - y = 2.

Answer

The graph of the equation x - y = 2 is a straight line passing through the points (0, -2), (2, 0), and (4, 2).

Ensure to use a ruler for drawing the straight line and label the axes and the line.

Example 3

Check whether (0, 2) and (2, 0) are solutions of the equation x + y = 2.

IGiven equation is x + y = 2.
IITo check if (0, 2) is a solution, substitute x = 0 and y = 2 into the L.H.S. of the equation:
IIIL.H.S. = x + y = 0 + 2 = 2.
IVSince L.H.S. = 2 and R.H.S. = 2, L.H.S. = R.H.S. Thus, (0, 2) is a solution.
VTo check if (2, 0) is a solution, substitute x = 2 and y = 0 into the L.H.S. of the equation:
VIL.H.S. = x + y = 2 + 0 = 2.
VIISince L.H.S. = 2 and R.H.S. = 2, L.H.S. = R.H.S. Thus, (2, 0) is a solution.

Answer

Both (0, 2) and (2, 0) are solutions of the equation x + y = 2.

Common mistakes

  • ✗Confusing a linear equation in one variable (e.g., 2x + 3 = 0) with a linear equation in two variables (e.g., 2x + y + 3 = 0).
  • ✗Incorrectly substituting values for x and y, leading to errors in finding solutions.
  • ✗Plotting points inaccurately on the Cartesian plane, which results in an incorrect graph.
  • ✗Drawing a curve instead of a straight line when graphing a linear equation.
  • ✗Not finding at least two distinct solutions before attempting to draw the graph, or not using a third point to verify collinearity.

Exam tips

  • ★Always write the linear equation in the standard form ax + by + c = 0 if it is not already in that form.
  • ★When finding solutions, choose simple integer values for one variable (like 0, 1, -1, 2) to simplify calculations for the other variable.
  • ★For graphing, always find at least three solutions. If these three points are not collinear, recheck your calculations for errors.
  • ★Use a sharp pencil and a ruler for drawing graphs to ensure neatness and accuracy. Label the axes (X-axis, Y-axis), the origin (O), and the line with its equation.
  • ★Clearly show all steps for finding solutions and plotting points, as marks are often awarded for working.

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