Question

How do you find slope and intercepts to graph $9x-5y=4$?

Answer 1

Hint: In the above question, we have been given an equation of a straight line, since it is linear in both x and y. The slope is defined as the derivative of y with respect to x. Therefore, on differentiating the given equation with respect to x, we can solve the resulting equation for $\dfrac{dy}{dx}$ to get the value of the slope. And the intercept is the point on which the graph of a function intersects the y-axis. Therefore, we will put $x=0$ in the given equation and on solving the resulting equation for y, we will obtain the value of the intercept.

Complete step-by-step answer:
The equation given in the above question is
$\Rightarrow 9x-5y=4$
Since the above equation is linear in both x and y, it is the equation of a straight line. Now, we know that the slope is equal to the derivative of y with respect to x. Therefore, we differentiate the above equation with respect to x to get
$\begin{align}
  & \Rightarrow \dfrac{d\left( 9x-5y \right)}{dx}=\dfrac{d\left( 4 \right)}{dx} \\
 & \Rightarrow 9-5\dfrac{dy}{dx}=0 \\
\end{align}$
Subtracting $9$ from both the sides, we get
$\begin{align}
  & \Rightarrow 9-5\dfrac{dy}{dx}-9=0-9 \\
 & \Rightarrow -5\dfrac{dy}{dx}=-9 \\
\end{align}$
Dividing both the sides by $-5$, we get
$\begin{align}
  & \Rightarrow \dfrac{dy}{dx}=\dfrac{-9}{-5} \\
 & \Rightarrow \dfrac{dy}{dx}=\dfrac{9}{5} \\
\end{align}$
Hence, the slope is equal to $\dfrac{9}{5}$.
Now, we know that the intercept is the point where the graph intersects the y-axis. Since the equation o y-axis is $x=0$, we substitute it in the given equation to get
$\begin{align}
  & \Rightarrow 9\left( 0 \right)-5y=4 \\
 & \Rightarrow -5y=4 \\
 & \Rightarrow y=-\dfrac{4}{5} \\
\end{align}$
Hence, the intercept is equal to $-\dfrac{4}{5}$. In decimal form, we can write it as -0.8.
We can observe these in the below graph.

Note: We can also easily determine the slope and intercept of the given equation using the by writing it in the slope-intercept form, which is given as $y=mx+c$. For this, we need to separate y on the LHS such that its coefficient is equal to one. Then the coefficient of x will be equal to the sloep and the constant term will be equal to the intercept.