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

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.

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 $\dfrac{{ - b}}{a}$.