Pair of Linear Equations in Two Variables — Class 10

Graphical method • Substitution • Elimination • Cross Multiplication • Practice
Jump to Practice

1. Introduction

A pair of linear equations in two variables is a system of two equations of the form:

a1x + b1y + c1 = 0
a2x + b2y + c2 = 0

The solution (x,y) is the point of intersection of the two lines represented by these equations.

2. Graphical Method

Plot both lines on a graph and find their intersection point (if any). Useful for visual understanding but less precise for exam answers unless coordinates are exact.

Example. Solve by graphing: x + y = 5 and x − y = 1. Intersection at (3,2).

Tip: Make a table of values for each equation (x → y) and plot two points per line to draw it accurately.

3. Substitution Method

From one equation, express one variable in terms of the other and substitute into the second equation to solve.

Solve: x + 2y = 7 and 3x − y = 4.

  1. From first: x = 7 − 2y.
  2. Substitute in second: 3(7−2y) − y = 4 ⇒ 21 −6y − y =4 ⇒ −7y = −17 ⇒ y = 17/7.
  3. x = 7 − 2(17/7) = (49 − 34)/7 = 15/7.

4. Elimination (Addition/Subtraction) Method

Multiply equations to make coefficients of one variable equal and add/subtract to eliminate it. Solve for the remaining variable and back-substitute.

Solve: 2x + 3y = 13 and 3x − 2y = 4.

  1. Multiply first by 3 and second by 2: 6x + 9y = 39 and 6x − 4y = 8.
  2. Subtract: 13y = 31 ⇒ y = 31/13 = 31/13 (simplify if possible).
  3. Back-substitute to find x.

5. Cross Multiplication Method

For system in standard form a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0, the solution (if unique) is given by:

x = \frac{b_1c_2 - b_2c_1}{a_1b_2 - a_2b_1}, y = \frac{c_1a_2 - c_2a_1}{a_1b_2 - a_2b_1}

Use carefully—ensure denominator (a1b2 − a2b1) ≠ 0.

6. Consistency of System

Let Δ = a1b2 − a2b1, Δx = b1c2 − b2c1, Δy = c1a2 − c2a1 (using standard form). Then:

  • If Δ ≠ 0 ⇒ unique solution (consistent and independent).
  • If Δ = 0 and Δx = 0 and Δy = 0 ⇒ infinitely many solutions (dependent).
  • If Δ = 0 and at least one of Δx, Δy ≠ 0 ⇒ no solution (inconsistent / parallel lines).

7. Solved Examples

Example 1. Solve 2x + y = 5 and x − y = 1 using elimination.

Add equations: 3x = 6 ⇒ x = 2. Then y = 1.

Example 2. Determine consistency: x + 2y + 3 = 0 and 2x + 4y + 7 = 0.

Second is not a multiple of first (constants differ), Δ = 0 but Δx or Δy ≠ 0 ⇒ no solution (parallel).

8. Practice Questions

  1. Solve: x + y = 9 and 2x − y = 4. (Answer toggle)
  2. Use substitution to solve: 3x + 4y = 10 and x − 2y = 1.
  3. Find conditions on k such that the system has infinite solutions: kx + 2y = 3 and 2kx + 4y = 6.
  4. Use cross multiplication to solve: 4x − 5y = 1 and 2x + 3y = 7.

9. Exam Tips

  • Choose elimination when coefficients are small integers and substitution when one variable is already isolated.
  • Check special cases: if coefficients are proportional, inspect constants to decide consistency.
  • Practice synthetic manipulation for speed in exams.