Question

Find the x and y-intercepts of the line 4x-y-7 = 0 by converting the given equation in double intercept form $\dfrac{x}{a}+\dfrac{y}{b}=1$
[a] $\dfrac{x}{\dfrac{4}{7}}-\dfrac{y}{7}=1,a=\dfrac{4}{7},b=-7$
[b] $\dfrac{x}{\dfrac{4}{7}}+\dfrac{y}{-7}=1,a=\dfrac{4}{7},b=-7$
[c] $\dfrac{x}{\dfrac{7}{4}}-\dfrac{y}{7}=1,a=\dfrac{7}{4},b=7$
[d] $\dfrac{x}{\dfrac{7}{4}}+\dfrac{y}{-7}=1,a=\dfrac{7}{4},b=-7$

Answer 1

Hint: First convert the equation in ax+by = c form. Then divide both sides by c. Make numerators independent of a and b by dividing numerators and denominators of individual fractions by a and b respectively.


Complete step-by-step solution -

We have 4x-y-7=0.
Adding 7 on both sides, we get
4x - y - 7 + 7 = 0 + 7
i.e. 4x - y = 7
Dividing both sides by 7, we get
$\begin{align}
  & \dfrac{4x}{7}-\dfrac{y}{7}=1 \\
 & \Rightarrow \dfrac{4x}{7}+\dfrac{-y}{7}=1 \\
\end{align}$
Dividing numerator and denominator of $\dfrac{4x}{7}$ by 4 and that of $\dfrac{-y}{7}$ by -1, we get
$\begin{align}
  & \dfrac{\dfrac{4x}{4}}{\dfrac{7}{4}}+\dfrac{\dfrac{-y}{-1}}{\dfrac{7}{-1}}=1 \\
 & \Rightarrow \dfrac{x}{\dfrac{7}{4}}+\dfrac{y}{-7}=1 \\
\end{align}$
Comparing with $\dfrac{x}{a}+\dfrac{y}{b}=1$, we get
$a=\dfrac{7}{4}$ and $b=-7$.
Hence option [d] is correct.

Note: Alternatively we can find x-intercept by putting y = 0 and y-intercept by putting x = 0 in the equation of the line.
If y = 0, we have
4x-(0)-7=0
i.e. 4x = 7
Dividing both sides by 4, we get
$\begin{align}
  & \dfrac{4x}{4}=\dfrac{7}{4} \\
 & \Rightarrow x=\dfrac{7}{4} \\
\end{align}$
Hence we have $a=\dfrac{7}{4}$
Similarly, if x = 0, we have
4(0)-y-7=0
-y-7=0
Adding 7 both sides, we get
-y-7+7 = 0+7
i.e. -y = 7
Multiplying by -1 on both sides, we get
-1(-y) = -1(7)
y = -7
Hence b = -7
Which is the same as obtained above.
Alternatively: For lines in general form Ax+By+C = 0, we have x-intercept = $\dfrac{-C}{A}$ and y-intercept = $\dfrac{-C}{B}$.
We have A = 4, B = -1 and C =-7. Using the above result, we have
x-intercept $=\dfrac{-C}{A}=\dfrac{-\left( -7 \right)}{4}=\dfrac{7}{4}$ and y-intercept $=\dfrac{-C}{B}=\dfrac{-\left( -7 \right)}{-1}=-7$