Question

How do you find the slope and intercept to graph $y=3x+1$?

Answer 1

The equation $y=3x+1$ is linear in both the variables $x$ and $y$, which means that the graph of the given equation is a line. For determining the slope of the given line, we need to differentiate the given equation with respect to $x$. And for determining the intercept of the graph, we have to substitute $x=0$ in the given equation. The value of $y$ which will be obtained by substituting $x=0$ in the given equation will be the required intercept of the graph.

Complete step-by-step solution:
The equation of the graph is given in the question as
$y=3x+1......(i)$
Since the equation is linear with respect to both the variables $x$ and $y$, so the graph of the given equation is a straight line, as shown below.

Now, we know that the slope of a curve at a point is given by the derivative $\dfrac{dy}{dx}$ of the curve at that point. So we differentiate both sides of the above equation with respect to $x$ to get
$\begin{aligned} & \Rightarrow \dfrac{dy}{dx}=\dfrac{d\left( 3x+1 \right)}{dx} \\\ & \Rightarrow \dfrac{dy}{dx}=\dfrac{d\left( 3x \right)}{dx}+\dfrac{d(1)}{dx} \\\ & \Rightarrow \dfrac{dy}{dx}=3+0 \\\ & \Rightarrow \dfrac{dy}{dx}=3 \\\ \end{aligned}$
Since the slope $m=\dfrac{dy}{dx}$, so the slope of the given graph is equal to $3$.
Now, the intercept of a graph is given by the point where it cuts the y-axis. We know that the x-coordinate is equal to zero on the y-axis. Therefore, we substitute $x=0$ in the given equation (i) to get

& \Rightarrow y=3\left( 0 \right)+1 \\\ & \Rightarrow y=1 \\\ \end{aligned}$$ So the intercept of the given graph is equal to $$1$$. **Hence, the slope and intercept to the graph $y=3x+1$ are equal to $3$ and $$1$$ respectively.** **Note:** We can observe that the equation $y=3x+1$ is written in the slope intercept form of $y=mx+c$. So on comparing the given equation with the slope intercept form, we will directly get the slope $m=3$ and the intercept $c=1$. This method will give the answer quickly in this case, as compared to that used in the above solution.