Class 9 Notes

Class 9 Mathematics Notes Chapter 4 (Chapter 4) – Examplar Problem (Englisha) Book

Philoid

Philoid

— 6 min read

Examplar Problem (Englisha)
Alright class, let's focus on Chapter 4: Linear Equations in Two Variables from your NCERT Exemplar book. This is a fundamental chapter, and understanding it well is crucial not just for your Class 9 exams but also forms the base for many concepts tested in government exams.

Chapter 4: Linear Equations in Two Variables - Detailed Notes for Exam Preparation

1. Introduction & Definition:

  • What is a Linear Equation in Two Variables?

    • An equation that can be written in the standard form:
      ax + by + c = 0
    • Here, 'x' and 'y' are the variables.
    • 'a', 'b', and 'c' are real numbers.
    • Crucially, 'a' and 'b' cannot both be zero simultaneously (a² + b² ≠ 0). If both were zero, we wouldn't have any variables left!
    • The term "linear" signifies that the highest power (degree) of the variables in the equation is 1.

  • Examples:

    • 2x + 3y - 5 = 0 (Here a=2, b=3, c=-5)
    • x - 4y = 7 (Can be written as x - 4y - 7 = 0; a=1, b=-4, c=-7)
    • y = 3x (Can be written as 3x - y + 0 = 0; a=3, b=-1, c=0)
    • 5x = 10 (Can be written as 5x + 0y - 10 = 0; a=5, b=0, c=-10. This is still a linear equation in two variables, even though 'y' isn't explicitly present in the simplified form).
    • -2y = 8 (Can be written as 0x - 2y - 8 = 0; a=0, b=-2, c=-8).

  • Non-Examples:

    • x² + y = 5 (Not linear because the power of x is 2)
    • xy + 3 = 0 (Not linear because the term xy has a combined degree of 2)
    • 1/x + y = 2 (Not linear because of the 1/x term)

2. Solutions of a Linear Equation in Two Variables:

  • What is a Solution?
    • 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., Left Hand Side (LHS) = Right Hand Side (RHS)).
  • Infinitely Many Solutions:
    • A key property: A linear equation in two variables has infinitely many solutions.
    • You can choose any real value for one variable (say 'x') and calculate the corresponding value for the other variable ('y') that satisfies the equation.
  • Finding Solutions:
    • Method:
      1. Rewrite the equation to express one variable in terms of the other (e.g., solve for y in terms of x, or vice-versa).
      2. Choose convenient values for one variable (often integers like 0, 1, -1, etc.).
      3. Substitute this value into the rearranged equation to find the corresponding value of the other variable.
      4. Write the solution as an ordered pair (x, y).
    • Example: Find two solutions for 2x + y = 6.
      1. Rearrange: y = 6 - 2x
      2. Let x = 0: y = 6 - 2(0) = 6. Solution: (0, 6)
      3. Let x = 1: y = 6 - 2(1) = 4. Solution: (1, 4)
      4. Let x = 3: y = 6 - 2(3) = 0. Solution: (3, 0)
        (You can find infinitely many more)
  • Checking a Solution: Substitute the x and y values from the ordered pair into the original equation. If LHS = RHS, it's a valid solution.
    • Example: Is (1, 4) a solution for 2x + y = 6?
      • LHS = 2(1) + 4 = 2 + 4 = 6
      • RHS = 6
      • Since LHS = RHS, yes, (1, 4) is a solution.

3. Graph of a Linear Equation in Two Variables:

  • The Graph is a Straight Line: The geometric representation of all the solutions (x, y) of a linear equation in two variables is always a straight line on the Cartesian plane.
  • Every point on the line is a solution.
  • Every solution corresponds to a point on the line.
  • How to Draw the Graph:
    1. Find at least two distinct solutions (ordered pairs) for the equation using the method described above.
    2. Plot these two points on the Cartesian coordinate system (graph paper).
    3. Draw a straight line passing through these two points. Extend the line in both directions with arrows.
    4. (Good Practice): Find a third solution and check if it lies on the same line. This helps verify your calculations.
  • Intercepts:
    • x-intercept: The point where the line crosses the x-axis. At this point, the y-coordinate is always 0. To find it, set y = 0 in the equation and solve for x. The point is (x, 0).
    • y-intercept: The point where the line crosses the y-axis. At this point, the x-coordinate is always 0. To find it, set x = 0 in the equation and solve for y. The point is (0, y).
    • Example: For 2x + y = 6:
      • x-intercept (set y=0): 2x + 0 = 6 => 2x = 6 => x = 3. Point: (3, 0).
      • y-intercept (set x=0): 2(0) + y = 6 => y = 6. Point: (0, 6).
        (These are the same points we found earlier)

4. Equations of Lines Parallel to the Axes:

  • Equation of the x-axis: y = 0 (Every point on the x-axis has a y-coordinate of 0).
  • Equation of the y-axis: x = 0 (Every point on the y-axis has an x-coordinate of 0).
  • Line Parallel to the y-axis:
    • The equation is of the form x = k (where 'k' is a constant).
    • This is a vertical line passing through the point (k, 0). All points on this line have their x-coordinate equal to 'k'.
    • Example: x = 3 is a line parallel to the y-axis, 3 units to its right.
  • Line Parallel to the x-axis:
    • The equation is of the form y = k (where 'k' is a constant).
    • This is a horizontal line passing through the point (0, k). All points on this line have their y-coordinate equal to 'k'.
    • Example: y = -2 is a line parallel to the x-axis, 2 units below it.
  • Representation as ax + by + c = 0:
    • 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 are special cases of linear equations in two variables where one coefficient (a or b) is zero.

5. Representing Word Problems:

  • Many real-life situations can be modelled using linear equations in two variables.
  • Steps:
    1. Read the problem carefully and identify the two unknown quantities.
    2. Assign variables (usually x and y) to these quantities. Clearly state what each variable represents.
    3. Translate the relationship or condition described in the problem into a mathematical equation involving x and y.
    • Example: "The cost of a notebook is twice the cost of a pen. Represent this statement as a linear equation in two variables."
      1. Unknowns: Cost of a notebook, Cost of a pen.
      2. Variables: Let x = cost of a notebook (in ₹), Let y = cost of a pen (in ₹).
      3. Equation: Cost of notebook = 2 * Cost of pen => x = 2y
      4. Standard form: x - 2y = 0 (or x - 2y + 0 = 0)

Multiple Choice Questions (MCQs)

  1. The standard form of a linear equation in two variables x and y is:
    (A) ax + b = 0, a ≠ 0
    (B) ax + by = c, a ≠ 0, b ≠ 0
    (C) ax + by + c = 0, where a, b, c are real numbers and a² + b² ≠ 0
    (D) y = mx + c

  2. The equation x = 5, in two variables, can be written as:
    (A) 1.x + 1.y - 5 = 0
    (B) 0.x + 1.y - 5 = 0
    (C) 1.x + 0.y - 5 = 0
    (D) 1.x + 0.y + 5 = 0

  3. Any point on the y-axis is of the form:
    (A) (x, 0)
    (B) (0, y)
    (C) (x, x)
    (D) (y, y)

  4. The graph of the linear equation 2x + 3y = 6 cuts the y-axis at the point:
    (A) (2, 0)
    (B) (0, 3)
    (C) (3, 0)
    (D) (0, 2)

  5. How many linear equations in x and y can be satisfied by x = 1 and y = 2?
    (A) Only one
    (B) Two
    (C) Infinitely many
    (D) Three

  6. The point of the form (a, a) always lies on:
    (A) x-axis
    (B) y-axis
    (C) The line y = x
    (D) The line x + y = 0

  7. If (2, 0) is a solution of the linear equation 2x + 3y = k, then the value of k is:
    (A) 4
    (B) 6
    (C) 5
    (D) 2

  8. The graph of the line x = -3 is a line parallel to the:
    (A) x-axis
    (B) y-axis
    (C) line y = x
    (D) line y = -x

  9. The equation y = 5 represents a line:
    (A) Parallel to the y-axis at a distance of 5 units from the origin
    (B) Parallel to the x-axis at a distance of 5 units from the origin
    (C) Making an intercept of 5 on the x-axis
    (D) Making an intercept of 5 on both axes

  10. The cost of 5 pencils is equal to the cost of 2 pens. If the cost of a pencil is ₹ x and the cost of a pen is ₹ y, the linear equation representing this statement is:
    (A) 5x + 2y = 0
    (B) 2x - 5y = 0
    (C) 5x - 2y = 0
    (D) 2x + 5y = 0

Answer Key for MCQs:

  1. (C)
  2. (C)
  3. (B)
  4. (D) [Set x=0 => 3y=6 => y=2. Point is (0, 2)]
  5. (C) [Infinitely many lines can pass through a single point]
  6. (C) [In y=x, the y-coordinate is always equal to the x-coordinate]
  7. (A) [Substitute x=2, y=0 => 2(2) + 3(0) = k => 4 + 0 = k => k=4]
  8. (B)
  9. (B)
  10. (C) [Cost of 5 pencils = 5x, Cost of 2 pens = 2y. Given 5x = 2y => 5x - 2y = 0]

Make sure you practice drawing graphs and finding solutions for various equations. Understanding the relationship between the algebraic form (the equation) and the geometric form (the line) is essential. Keep revising these concepts!

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