Problem Solving

A given line \(L\) has an equation \(3x+4y=5.\) Which of the following is the equation of the line which does not intersect the above line?

(A) \(4x + 3y = 5\)
(B) \(3x + 4y = 10\)
(C) \(3x + 5y = 5\)
(D) \(3x + 5y = 3\)
(E) \(3x – 4y = 5\)

[spoiler]OA=B[/spoiler]

Source: Veritas Prep

2 lines will not intercept when they are parallel i.e they will have the same slope but different intercept on the y-axis
$$Expres\sin g\ the\ given\ equation\ in\ terms\ of\ y=mx+c$$
$$4y=-3x+5$$
$$y=\frac{-3x}{4}+\frac{5}{4}$$
$$m\left(slope\right)=\frac{-3}{4}\ and\ y\ \left(intercept\right)=\frac{5}{4}$$
$$testing\ all\ available\ options$$
$$\left(A\right)=>y=\frac{-4}{3}x+\frac{5}{3};\ m=\frac{-4}{3}and\ y\ intercept\ =\frac{5}{3}$$ $$\left(C\right)=>y=\frac{-3}{5}x+1;\ m=\frac{-3}{5}and\ y\ intercept\ =1$$
$$\left(D\right)=>y=\frac{-3}{5}x+\frac{3}{5};\ m=\frac{-3}{5}and\ y\ intercept\ =\frac{3}{5}$$
$$\left(E\right)=>y=\frac{3}{4}x+\frac{5}{4};\ m=+\frac{3}{4}and\ y\ intercept\ =\frac{5}{4}$$
$$But\ for\ option\ B=>y=\frac{-3}{4}x+\frac{10}{4}$$
$$=>y=\frac{-3}{4}x+\frac{5}{2};\ m=\frac{-3}{4}and\ y\ intercept=\frac{5}{2}$$
$$This\ gives\ same\ slope\ but\ different\ y\ intercept\ compared\ with\ the\ equation\ provided\ $$
$$so\ it\ can\ never\ inter\sec t$$
$$Answer\ =\ B$$