Question

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

Answer 1

Step 1: Find the $x-$intercept and $\mathrm{y}-$ intercept of given line

Given line is, $3\mathrm{x}+4\mathrm{y}=24$.

We will find the intercept of line by using the formula of intercept form.

The standard double-intercept form of a straight line is,

$\frac{x}{a}+\frac{y}{b}=1$

Here, $a$ is the $x-$intercept and $\mathrm{b}$ is $\mathrm{y}-$ intercept.

$\Rightarrow$ $3\mathrm{x}+4\mathrm{y}=24$

Divide both sides by $24$.

$\Rightarrow$ $\frac{3\mathrm{x}}{24}+\frac{4\mathrm{y}}{24}=\frac{24}{24}$

$\Rightarrow$ $\frac{\mathrm{x}}{8}+\frac{\mathrm{y}}{6}=1$

$\Rightarrow$ $x-$ intercept$=a=8$

$\Rightarrow$ $\mathrm{y}-$ intercept$=b=6$

Step 2 : Find the $x-$ intercept and $\mathrm{y}-$ intercept of line $L$

$x-$ intercept of $L$ is double of given line

Let $a_1$ be the $x-$ intercept of $L$

$\Rightarrow$ $a_1=2a$

$\Rightarrow$$a_1=2\times 8=16$

$\mathrm{y}-$ intercept of line $L$ is half of given line

Let $b_1$be the $\mathrm{y}-$intercept of line $L$

$\Rightarrow$$b_1=\frac{b}{2}$

$\Rightarrow$$b_1=\frac{6}{2}=3$

Step 3: Using double intercept form write the equation of line $L$

Apply double intercept form,

$\Rightarrow$ $\frac{x}{a_1}+\frac{y}{b_1}=1$

$\Rightarrow$ $\frac{x}{16}+\frac{y}{3}=1$

$\Rightarrow$ $3x+16y=48$

$\Rightarrow$$3x+16y-48=0$

Step 4: Find slope of line $L$

$3x+16y-48=0$ is the equation of line $L$

Comparing this equation with standard form of equation of straight line $ax+by+c=0$ we get

$a=3,b=16,c=-48$

The slope of the line represented by $ax+by+c=0$ is $m=\frac{-a}{b}$

Hence, slope of line $L$ is given by

$m=\frac{-a}{b}$

$\Rightarrow$$m=\frac{-3}{16}$

Hence, the slope of line is $\frac{-3}{16}$.