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
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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 \[\dfrac{{ - b}}{a}\].