Question

If the x-intercept of some line \[L\] is double as that of the line, \[3x + 4y = 24\] and the y-intercept of \[L\] is half as that of the same line, then the slope of \[L\] is

Answer 1

Hint: First of all, find the intercepts of the given line by converting it into line intercept form and then find the intercepts of the line \[L\] by using the given condition. Then find the line equation of line \[L\] and find its slope.

Complete step-by-step answer:
Given the x-intercept of some line \[L\] is double as that of the line, \[3x + 4y = 24\] and the y-intercept of the same line.
Converting the line \[3x + 4y = 24\], into intercept form we get
\[
   \Rightarrow 3x + 4y = 24 \\
   \Rightarrow \dfrac{{3x + 4y}}{{24}} = \dfrac{{24}}{{24}} \\
   \Rightarrow \dfrac{{3x}}{{24}} + \dfrac{{4y}}{{24}} = 1 \\
  \therefore \dfrac{x}{8} + \dfrac{y}{6} = 1 \\
\]
We know that for the line intercept form \[\dfrac{x}{a} + \dfrac{y}{b} = 1\], the x-intercept is \[a\] and the y-intercept is \[b\].
So, for the line \[3x + 4y = 24\], x-intercept is \[8\] and y-intercept is \[6\].
Hence x-intercept of line \[L = 2\left( 8 \right) = 16\]
            y-intercept of line \[L = \dfrac{6}{2} = 3\]
Thus, the line intercept form of line \[L\] is given by \[\dfrac{x}{{16}} + \dfrac{y}{3} = 1\].
We know that for the line intercept form \[\dfrac{x}{a} + \dfrac{y}{b} = 1\], the slope is given by \[\dfrac{{ - b}}{a}\].
So, slope of the line \[L = \dfrac{x}{{16}} + \dfrac{y}{3} = 1\] is \[\dfrac{{ - 3}}{{16}}\].
Thus, the slope of the line \[L\] is \[\dfrac{{ - 3}}{{16}}\].

Note: The x-intercept is where a line crosses the x-axis and y-intercept is the point where the line crosses the y-axis. For the line intercept form \[\dfrac{x}{a} + \dfrac{y}{b} = 1\], the x-intercept is \[a\] and the y-intercept is \[b\]. For the line intercept form \[\dfrac{x}{a} + \dfrac{y}{b} = 1\], the slope is given by \[\dfrac{{ - b}}{a}\].