Graph the two lines $y = 2 x + 1$ and $y = - x + 4$ and find their point of intersection. State whether the lines are parallel, perpendicular, or neither.

hard 4 marks Class 9 · Mathematics · Linear Equations in Two Variables

Answer

Intersection at

(1, 3)

; the lines are neither parallel nor perpendicular.

The correct answer is Intersection at $(1, 3)$ ; the lines are neither parallel nor perpendicular. — 2. Solve for $x$ : $2 x + 1 = - x + 4 \Rightarrow 3 x = 3 \Rightarrow x = 1$ . 3. Substitute $x = 1$ into one of the equations to find $y$ : $y = 2 \cdot 1 + 1 = 3$ (or $y = - 1 + 4 = 3$ ). 4. The point of intersection is $(1, 3)$ . 5. The slopes of the lines are $m_{1} = 2$ and $m_{2} = - 1$ . Since $m_{1} m_{2} = - 2 \neq = - 1$ and $m_{1} \neq = m_{2}$ , the lines are neither perpendicular nor parallel.

Step-by-step solution

1. Set the two equations equal to find intersection: $2 x + 1 = - x + 4$ . This equates $y$ values because at intersection both $x$ and $y$ satisfy both equations.
2. Solve for $x$ : $2 x + 1 = - x + 4 \Rightarrow 3 x = 3 \Rightarrow x = 1$ . 3. Substitute $x = 1$ into one of the equations to find $y$ : $y = 2 \cdot 1 + 1 = 3$ (or $y = - 1 + 4 = 3$ ). 4. The point of intersection is $(1, 3)$ . 5. The slopes of the lines are $m_{1} = 2$ and $m_{2} = - 1$ . Since $m_{1} m_{2} = - 2 \neq = - 1$ and $m_{1} \neq = m_{2}$ , the lines are neither perpendicular nor parallel.

Common mistake: Assuming the lines are parallel because they appear to cross at a single point

Concept tested: Find intersection of two linear equations

Published 13 October 2026 · How this solution was prepared and checked

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Graph the two lines y=2x+1 and y=−x+4 and find their point of intersection. State whether the lines are parallel, perpendicular, or neither.

Step-by-step solution

Graph the two lines $y = 2 x + 1$ and $y = - x + 4$ and find their point of intersection. State whether the lines are parallel, perpendicular, or neither.