Class 10 Mathematics Notes Chapter 3 (Pair of linear equations in two variables) – Mathematics Book

Dear students, let's read notes of Chapter 3: Pair of Linear Equations in Two Variables. This is a fundamental chapter, and understanding it well is crucial not just for your Class 10 exams but also forms the basis for many concepts you'll encounter in competitive government exams.

Chapter 3: Pair of Linear Equations in Two Variables - Detailed Notes

1. Introduction

• Linear Equation in Two Variables: An equation that 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. 'x' and 'y' are the variables.
  • Example: 2x + 3y - 5 = 0
• Solution of a Linear Equation: A pair of values (one for x and one for y) which makes the two sides of the equation equal. A single linear equation in two variables has infinitely many solutions. Graphically, each solution represents a point on the line representing the equation.
• Pair of Linear Equations in Two Variables: Two linear equations involving the same two variables, x and y. The general form is:
  • a₁x + b₁y + c₁ = 0
  • a₂x + b₂y + c₂ = 0
    (where a₁, b₁, c₁, a₂, b₂, c₂ are real numbers, and a₁²+b₁² ≠ 0, a₂²+b₂² ≠ 0).
• Solution of a Pair of Linear Equations: A pair of values (x, y) that satisfies both equations simultaneously. This represents the point of intersection of the two lines represented by the equations.

2. Graphical Representation of a Pair of Linear Equations

When we draw the graphs of two linear equations in two variables on the same coordinate plane, we get two straight lines. There are three possibilities:

• (i) Intersecting Lines: The lines cross at exactly one point.

  • Algebraic Interpretation: The pair of equations has a unique solution.
  • Condition: a₁/a₂ ≠ b₁/b₂
  • Consistency: The system is called consistent.

• (ii) Parallel Lines: The lines never cross. They maintain a constant distance apart.

  • Algebraic Interpretation: The pair of equations has no solution.
  • Condition: a₁/a₂ = b₁/b₂ ≠ c₁/c₂
  • Consistency: The system is called inconsistent.

• (iii) Coincident Lines: One line lies exactly on top of the other. Essentially, they are the same line.

  • Algebraic Interpretation: The pair of equations has infinitely many solutions. (Every point on the line is a solution).
  • Condition: a₁/a₂ = b₁/b₂ = c₁/c₂
  • Consistency: The system is called consistent (and dependent).

Summary Table (Very Important for Exams):

3. Algebraic Methods of Solving a Pair of Linear Equations

While the graphical method gives a visual understanding, algebraic methods provide exact solutions.

• (i) Substitution Method:

  1. Express one variable (say y) in terms of the other variable (x) from one of the equations.
  2. Substitute this expression for y into the other equation. This results in a linear equation in one variable (x).
  3. Solve this equation to find the value of x.
  4. Substitute the found value of x back into the expression from step 1 to find the value of y.

• (ii) Elimination Method:

  1. Multiply one or both equations by suitable non-zero constants to make the coefficients of one variable (either x or y) numerically equal.
  2. Add or subtract the modified equations to eliminate the variable with equal coefficients. This results in a linear equation in one variable.
  3. Solve this equation to find the value of the remaining variable.
  4. Substitute this value back into either of the original equations to find the value of the other variable.

• (iii) Cross-Multiplication Method: (Useful for quick calculations if you remember the pattern)
  For the pair of equations:
  a₁x + b₁y + c₁ = 0
  a₂x + b₂y + c₂ = 0
  The solution is given by:
  x / (b₁c₂ - b₂c₁) = y / (c₁a₂ - c₂a₁) = 1 / (a₁b₂ - a₂b₁)

  • How to remember: Write the coefficients in the following pattern and cross-multiply:

          x          y          1
       b₁  c₁      a₁  b₁
       b₂  c₂      a₂  b₂
    

    x = (b₁c₂ - b₂c₁) / (a₁b₂ - a₂b₁)
    y = (c₁a₂ - c₂a₁) / (a₁b₂ - a₂b₁)

  • Important Note: This method works only when (a₁b₂ - a₂b₁) ≠ 0, which is the condition for a unique solution (a₁/a₂ ≠ b₁/b₂). If (a₁b₂ - a₂b₁) = 0, the system either has no solution or infinitely many solutions.

4. Equations Reducible to a Pair of Linear Equations in Two Variables

Sometimes, equations are not linear initially but can be transformed into a linear pair by making suitable substitutions.

• Example: Solve 2/x + 3/y = 13 and 5/x - 4/y = -2.
  • Substitute: Let 1/x = u and 1/y = v.
  • The equations become: 2u + 3v = 13 and 5u - 4v = -2.
  • Solve this linear pair for u and v using any algebraic method.
  • Once you find u and v, find x and y using x = 1/u and y = 1/v.

5. Word Problems

This chapter is heavily used in solving word problems involving two unknown quantities.

• Steps:
  1. Read the problem carefully and identify the two unknown quantities. Assign variables (e.g., x and y) to them.
  2. Translate the statements/conditions given in the problem into two distinct linear equations in terms of x and y.
  3. Solve the pair of linear equations using the most convenient method (Substitution, Elimination, or Cross-Multiplication).
  4. Check if the solution satisfies the conditions of the problem.
  5. Write the answer in the context of the problem.
• Common Types: Problems based on ages, numbers, fractions, speed-distance-time, geometry (angles, area), fixed charges and per-unit charges, etc.

Key Takeaways for Government Exams:

• Focus heavily on the conditions for consistency (the table comparing ratios). Many MCQs are directly based on this.
• Be proficient in at least Elimination and Substitution methods. Cross-multiplication is faster if you're comfortable with it.
• Practice word problems to quickly form equations from given information.
• Understand the graphical interpretation – what parallel, intersecting, and coincident lines mean in terms of solutions.
• Recognize equations reducible to linear form.

Multiple Choice Questions (MCQs)

1. The pair of equations y = 0 and y = -5 has:
   (a) One solution
   (b) Two solutions
   (c) Infinitely many solutions
   (d) No solution

2. If the lines given by 3x + 2ky = 2 and 2x + 5y + 1 = 0 are parallel, then the value of k is:
   (a) -5/4
   (b) 2/5
   (c) 15/4
   (d) 3/2

3. The pair of equations x + 2y + 5 = 0 and -3x - 6y + 1 = 0 have:
   (a) A unique solution
   (b) Exactly two solutions
   (c) Infinitely many solutions
   (d) No solution

4. For what value of k, do the equations 2x - 3y + 10 = 0 and 3x + ky + 15 = 0 represent coincident lines?
   (a) -9/2
   (b) -11
   (c) 9/2
   (d) -7

5. The solution of the equations x - y = 2 and x + y = 4 is:
   (a) x = 3, y = 1
   (b) x = 3, y = -1
   (c) x = -3, y = 1
   (d) x = -3, y = -1

6. The graphical representation of the pair of equations x + 2y - 4 = 0 and 2x + 4y - 12 = 0 is:
   (a) Intersecting lines
   (b) Parallel lines
   (c) Coincident lines
   (d) Perpendicular lines

7. If a pair of linear equations is consistent, then the lines will be:
   (a) Parallel
   (b) Always coincident
   (c) Intersecting or coincident
   (d) Always intersecting

8. The sum of two numbers is 35 and their difference is 13. The numbers are:
   (a) 24 and 11
   (b) 20 and 15
   (c) 25 and 10
   (d) 22 and 13

9. If we substitute u = 1/x and v = 1/y in the equations 2/x + 3/y = 6 and 1/x + 1/y = 5/6, the transformed linear equations are:
   (a) 2u + 3v = 6, u + v = 5/6
   (b) 3u + 2v = 6, u + v = 5/6
   (c) 2u + 3v = 1/6, u + v = 6/5
   (d) u/2 + v/3 = 6, u + v = 5/6

10. The pair of equations 5x - 15y = 8 and 3x - 9y = 24/5 has:
    (a) One solution
    (b) Two solutions
    (c) Infinitely many solutions
    (d) No solution

Answer Key for MCQs:

1. (d) No solution (Both lines are horizontal and parallel: y=0 is the x-axis, y=-5 is a line parallel to the x-axis).
2. (c) 15/4 (For parallel lines, a₁/a₂ = b₁/b₂ ≠ c₁/c₂. So, 3/2 = 2k/5 => 15 = 4k => k = 15/4).
3. (d) No solution (a₁/a₂ = 1/-3; b₁/b₂ = 2/-6 = 1/-3; c₁/c₂ = 5/1. Since a₁/a₂ = b₁/b₂ ≠ c₁/c₂, the lines are parallel).
4. (a) -9/2 (For coincident lines, a₁/a₂ = b₁/b₂ = c₁/c₂. So, 2/3 = -3/k = 10/15. From 2/3 = -3/k => 2k = -9 => k = -9/2).
5. (a) x = 3, y = 1 (Adding the two equations gives 2x = 6 => x = 3. Substituting x=3 in x-y=2 gives 3-y=2 => y=1).
6. (b) Parallel lines (a₁/a₂ = 1/2; b₁/b₂ = 2/4 = 1/2; c₁/c₂ = -4/-12 = 1/3. Since a₁/a₂ = b₁/b₂ ≠ c₁/c₂, the lines are parallel).
7. (c) Intersecting or coincident (Consistent means at least one solution exists, which happens for intersecting or coincident lines).
8. (a) 24 and 11 (Let numbers be x and y. x + y = 35, x - y = 13. Adding gives 2x = 48 => x = 24. Then 24 + y = 35 => y = 11).
9. (a) 2u + 3v = 6, u + v = 5/6 (Direct substitution of 1/x = u and 1/y = v).
10. (c) Infinitely many solutions (Equation 2: 3x - 9y = 24/5. Divide by 3: x - 3y = 8/5. Multiply by 5: 5x - 15y = 8. This is identical to Equation 1. So, a₁/a₂ = 5/3; b₁/b₂ = -15/-9 = 5/3; c₁/c₂ = 8/(24/5) = 8 * 5 / 24 = 40/24 = 5/3. Since a₁/a₂ = b₁/b₂ = c₁/c₂, the lines are coincident).

Study these notes thoroughly, practice solving problems using all methods, and pay special attention to the conditions for consistency. Good luck with your preparation!