The lines $x - y + 1 = 0$ and $3 x + 2 y - 12 = 0$ are drawn on the grid (1 square = 1 unit). Read the vertices of the shaded triangle formed by these lines and the $x$ -axis.

Class 10Pair of Linear Equations in Two VariablesAlgebra — Systems of Equations

A(0, 1), (0, 6) and (2, 3)
B(−1, 0), (4, 0) and (2, 3)
C(1, 0), (4, 0) and (2, 3)
D(−1, 0), (4, 0) and (3, 2)

Step-by-Step Solution

x - y + 1 = 0

meets the

x

-axis

(y = 0)

(- 1, 0)

;

3 x + 2 y - 12 = 0

meets it at

(4, 0)

. The two lines meet at

(2, 3) : 2 - 3 + 1 = 0

and

6 + 6 - 12 = 0

Answer: B) (−1, 0), (4, 0) and (2, 3)

Key idea: Three geometric possibilities

Two lines in a plane can:

1. **Intersect at a single point** → exactly ONE solution. (Independent, consistent.)
2. **Be parallel and distinct** → NO solution. (Inconsistent.)
3. **Coincide (be the same line)** → INFINITELY many solutions. (Dependent, consistent.)

These three cases are decided by comparing the coefficient ratios:

| Condition on a₁/a₂, b₁/b₂, c₁/c₂ | Type | # solutions |
|---|---|---|
| a₁/a₂ ≠ b₁/b₂ | Lines intersect | One |
| a₁/a₂ = b₁/b₂ ≠ c₁/c₂ | Parallel | None |
| a₁/a₂ = b₁/b₂ = c₁/c₂ | Coincident | Infinitely many |

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The lines x−y+1=0 and 3x+2y−12=0 are drawn on the grid (1 square = 1 unit). Read the vertices of the shaded triangle formed by these lines and the x-axis.

Step-by-Step Solution

Key idea: Three geometric possibilities

Master this chapter in 30 minutes a day.

More Class 10 Pair of Linear Equations in Two Variables practice

The lines $x - y + 1 = 0$ and $3 x + 2 y - 12 = 0$ are drawn on the grid (1 square = 1 unit). Read the vertices of the shaded triangle formed by these lines and the $x$ -axis.