find the equation of the line through the intersection of the lines x+2y-3=0 and 4x -y+7=0 and which is parallel - Maths - Straight Lines

Question

find the equation of the line through the intersection of the
lines x+2y-3=0 and 4x -y+7=0 and which is parallel to 5x +4y
-20=0

Priyanka Kedia · Accepted Answer

The equation of given lines are,

$x + 2 y - 3 = 0 ...... (1) 4 x - y + 7 = 0 ...... (2)$

The equation of line passing through the intersection of given line (1) and (2) is as,

$(x + 2 y - 3) + λ (4 x - y + 7) =0 ...... (3) \Rightarrow (4 λ + 1) x + (2 - λ) y + (7 λ - 3) = 0 ...... (4) \Rightarrow (2 - λ) y = - (4 λ + 1) x + (7 λ - 3) \Rightarrow y = - \frac{4 λ + 1}{2 - λ} + \frac{7 λ - 3}{2 - λ} T h u s, s l o p e o f l i n e = - \frac{4 λ + 1}{2 - λ}$

Given, this line (4) is parallel to given line,

$5 x + 4 y - 20 = 0 ...... (5) \Rightarrow 4 y = - 5 x + 20 \Rightarrow y = - \frac{5}{4} x + 5 T h u s, S l o p e o f l i n e = - \frac{5}{4}$

So, Slope of line (5) = Slope of line (4)

$- \frac{5}{4} = - \frac{4 λ + 1}{2 - λ} \Rightarrow \frac{5}{4} = \frac{4 λ + 1}{2 - λ} \Rightarrow 16 λ + 4 = 10 - 5 λ \Rightarrow 21 λ = 6 \Rightarrow λ = \frac{2}{7}$

Substitute value of $λ$ in equation (3), we get,

$(x + 2 y - 3) + \frac{2}{7} (4 x - y + 7) =0 \Rightarrow 7 (x + 2 y - 3) + 2 (4 x - y + 7) = 0 \Rightarrow 7 x + 14 y - 21 + 8 x - 2 y + 14 = 0 \Rightarrow 15 x + 12 y - 7 = 0$

Hence, equation of line through the intersection of line (1) and (2) and parallel to line (5) is 15x+12y-7=0.

find the equation of the line through the intersection of the lines x+2y-3=0 and 4x -y+7=0 and which is parallel to 5x +4y -20=0.