Class 9 Mathematics Notes Chapter 4 (Linear Equations in Two Variables) – Mathematics Book

Alright class, let's get straight into Chapter 4: Linear Equations in Two Variables. This is a fundamental chapter, not just for your Class 9 exams, but it lays the groundwork for many concepts you'll encounter in higher mathematics and quantitative sections of government exams. Pay close attention!

Chapter 4: Linear Equations in Two Variables - Detailed Notes

1. What is a Linear Equation in Two Variables?

• Definition: An equation that can be written in the form:
  ax + by + c = 0
  where:

  • x and y are variables.
  • a, b, and c are real numbers.
  • Crucially, a and b are not both zero (i.e., a² + b² ≠ 0).

• Why "Linear"? Because when you graph this equation on a Cartesian plane, it always forms a straight line.

• Why "Two Variables"? Because it involves two distinct variables (usually x and y).

• Example: 2x + 3y - 5 = 0 is a linear equation in two variables where a=2, b=3, c=-5.

• Non-Example: x² + y = 5 (not linear because of x²), xy = 4 (not linear because variables are multiplied), x + 1/y = 3 (not linear because of 1/y).

2. Standard Form:

• The standard form ax + by + c = 0 is important for identifying coefficients and the constant term.
• You might be given equations in other forms, like y = mx + c (slope-intercept form, which you'll study more later) or 2x = 5y - 1. You should be able to rearrange them into the standard form.
  • Example: Rearrange y = -3x + 7 to standard form.
    Add 3x to both sides: 3x + y = 7
    Subtract 7 from both sides: 3x + y - 7 = 0. Here, a=3, b=1, c=-7.
  • Example: Rearrange x/2 - y/3 = 1 to standard form.
    Multiply by LCM(2,3) = 6: 3x - 2y = 6
    Subtract 6: 3x - 2y - 6 = 0. Here, a=3, b=-2, c=-6.

3. Solutions of a Linear Equation in Two Variables:

• Definition: A solution is a pair of values, one for x and one for y, (written as an ordered pair (x, y)), that makes the equation true (i.e., LHS = RHS).
• Key Property: A linear equation in two variables has infinitely many solutions.
• Finding Solutions: You can find solutions by:
  1. Choosing any value for one variable (say x).
  2. Substituting this value into the equation.
  3. Solving the resulting linear equation in one variable to find the value of the other variable (y).
• Example: Find four solutions for x + 2y = 6.
  • Let x = 0: 0 + 2y = 6 => 2y = 6 => y = 3. Solution: (0, 3)
  • Let y = 0: x + 2(0) = 6 => x = 6. Solution: (6, 0)
  • Let x = 2: 2 + 2y = 6 => 2y = 4 => y = 2. Solution: (2, 2)
  • Let y = 1: x + 2(1) = 6 => x + 2 = 6 => x = 4. Solution: (4, 1)
• Checking a Solution: To check if a given point (p, q) is a solution, substitute x=p and y=q into the equation. If the equation holds true, it is a solution.
  • Example: Is (1, 2) a solution for 3x - y = 1?
    Substitute: 3(1) - (2) = 3 - 2 = 1. Since 1 = 1 (RHS), yes, (1, 2) is a solution.

4. Graph of a Linear Equation in Two Variables:

• Representation: The graph is always a straight line.
• Connection to Solutions:
  • Every point (x, y) lying on the line is a solution to the equation.
  • Every solution (x, y) of the equation corresponds to a point on the line.
• How to Graph:
  1. Find at least two solutions (ordered pairs) for the equation. Finding a third solution is recommended as a check – if all three points don't lie on the same straight line, you've made a calculation error.
  2. Plot these points on a Cartesian coordinate plane.
  3. Draw a straight line passing through these points. Extend the line in both directions with arrows to indicate it continues infinitely.
  4. Label the line with the equation.
• Example: Graph x + y = 4.
  • Solutions: (0, 4), (4, 0), (2, 2).
  • Plot these points and draw the line passing through them.

5. Special Cases: Equations of Lines Parallel to Axes and Axes Themselves:

• Equation of the x-axis: y = 0 (Because every point on the x-axis has a y-coordinate of 0).

• Equation of the y-axis: x = 0 (Because every point on the y-axis has an x-coordinate of 0).

• Lines Parallel to the y-axis: Equations of the form x = k (where k is a constant). This is a vertical line passing through the point (k, 0).

  • Example: x = 3 is a line parallel to the y-axis, 3 units to its right. Every point on this line has an x-coordinate of 3, e.g., (3, 0), (3, 2), (3, -5).

• Lines Parallel to the x-axis: Equations of the form y = k (where k is a constant). This is a horizontal line passing through the point (0, k).

  • Example: y = -2 is a line parallel to the x-axis, 2 units below it. Every point on this line has a y-coordinate of -2, e.g., (0, -2), (4, -2), (-1, -2).

• Representing x=k and y=k as Linear Equations in Two Variables:

  • x = k can be written as 1.x + 0.y - k = 0. Here a=1, b=0, c=-k.
  • y = k can be written as 0.x + 1.y - k = 0. Here a=0, b=1, c=-k.
    (This shows they fit the ax + by + c = 0 definition, remembering that a and b cannot both be zero).

6. Lines Passing Through the Origin:

• If the constant term c in ax + by + c = 0 is zero, the equation becomes ax + by = 0.
• Such lines always pass through the origin (0, 0), because substituting x=0 and y=0 gives a(0) + b(0) = 0, which is always true.
• Example: 2x - y = 0. Solutions include (0, 0), (1, 2), (2, 4). The graph passes through the origin.

Relevance for Government Exams:

• Foundation: This chapter is the basis for understanding systems of linear equations (solving two or more equations simultaneously), which is a very common topic.
• Coordinate Geometry: It's intrinsically linked to coordinate geometry, plotting points, and understanding lines.
• Data Interpretation: Graphs of linear equations can represent real-world relationships (e.g., cost vs quantity, distance vs time at constant speed). Being able to interpret these graphs is crucial.
• Problem Solving: Word problems can often be modelled using linear equations in two variables.

Multiple Choice Questions (MCQs)

Here are 10 MCQs to test your understanding. These are typical of the level you might find in government exams based on this foundational topic.

1. The standard form of a linear equation in two variables x and y is:
a) ax + b = 0
b) ax + by = c
c) ax + by + c = 0, where a, b are not both zero
d) y = mx + c

2. The graph of the equation 2x + 5 = 0 is a line:
a) Parallel to the x-axis
b) Parallel to the y-axis
c) Passing through the origin
d) None of the above

3. How many solutions does the linear equation 3x - 7y = 10 have?
a) One unique solution
b) Two solutions
c) Infinitely many solutions
d) No solution

4. The point (2, -1) is a solution for which of the following equations?
a) x + y = 3
b) 2x - y = 5
c) x - 2y = 0
d) 3x + y = 4

5. The equation of the x-axis is:
a) x = 0
b) y = 0
c) x = k
d) y = k

6. If the point (3, 4) lies on the graph of the equation 3y = ax + 7, then the value of a is:
a) 2/3
b) 5/3
c) 1
d) 7/3

7. The graph of y = 5x is a line which:
a) Is parallel to the x-axis
b) Is parallel to the y-axis
c) Passes through the origin
d) Intersects the y-axis at (0, 5)

8. Expressing y = -5 as a linear equation in two variables in standard form gives:
a) 1.x + 0.y + 5 = 0
b) 0.x + 1.y - 5 = 0
c) 0.x + 1.y + 5 = 0
d) 1.x + 1.y + 5 = 0

9. Any point on the line y = x is of the form:
a) (a, -a)
b) (0, a)
c) (a, 0)
d) (a, a)

10. The graph of the linear equation 2x + 3y = 6 cuts the y-axis at the point:
a) (3, 0)
b) (0, 3)
c) (2, 0)
d) (0, 2)

Answer Key for MCQs:

1. c) (Definition of standard form with the condition on a and b)
2. b) (2x = -5 => x = -5/2. This is of the form x = k, which is parallel to the y-axis)
3. c) (A fundamental property of linear equations in two variables)
4. b) (Substitute x=2, y=-1 into each option. Only 2(2) - (-1) = 4 + 1 = 5 holds true)
5. b) (All points on the x-axis have y-coordinate 0)
6. b) (Substitute x=3, y=4 into 3y = ax + 7: 3(4) = a(3) + 7 => 12 = 3a + 7 => 5 = 3a => a = 5/3)
7. c) (The equation is of the form ax + by = 0 (since 5x - y = 0), so it passes through the origin (0,0))
8. c) (y = -5 => y + 5 = 0. In two variables: 0.x + 1.y + 5 = 0)
9. d) (On the line y=x, the y-coordinate is always equal to the x-coordinate)
10. d) (To find where it cuts the y-axis, put x=0. 2(0) + 3y = 6 => 3y = 6 => y = 2. The point is (0, 2))

Study these notes carefully and practice solving various problems related to finding solutions, graphing, and identifying properties of these equations. Good luck with your preparation!