Question

How do you find the slope and intercept of $3x+7=y$?

Answer 1

In this problem we need to calculate the slope and intercept of the given line $3x+7=y$. We know that slope is the ratio of variation in $y$ for unit variation in $x$. So, we will calculate the values of $y$ where $x=0$, $x=1$ by substituting those values in the given equation and simplifying them. Now we will take the difference between the values of $y$ and we will consider this as the slope of the given equation. Now we will take the value of $y$ whereas the intercept of the given equation.

Complete step-by-step solution:
Given equation, $3x+7=y$.
Substituting the value of $x=0$ in the above equation, then we will get
$\begin{aligned} & y=3\left( 0 \right)+7 \\\ & \Rightarrow y=7 \\\ \end{aligned}$
Substituting the value of $x=1$ in the above equation, then we will get
$\begin{aligned} & y=3\left( 1 \right)+7 \\\ & \Rightarrow y=10 \\\ \end{aligned}$
Now the slope of the given equation is the variation of the $y$ for unit variation in $x$. Mathematically we can write it as
$m=\dfrac{{{y}_{2}}-{{y}_{1}}}{{{x}_{2}}-{{x}_{1}}}$
Substituting the above calculated values in this equation, then we will get
$\begin{aligned} & m=\dfrac{10-7}{1-0} \\\ & \Rightarrow m=3 \\\ \end{aligned}$
Hence the slope of the given equation is $m=3$.
Now the intercept of the given line is the value of $y$ where $x=0$. We have already calculated the value of $y$ where $x=0$ which is nothing but $y=7$.
Hence the intercept of the given equation is $c=7$.

Note: In this problem we have calculated the slope and intercept by calculating the values of $y$. We can also calculate the slope and intercept without calculating the values of $y$. We have the slope intercept form of a line as $y=mx+c$ where $m$ is the slope of the line and $c$ is the $y$- intercept. So, we will compare the given equation with $y=mx+c$, then we will get
$m=3$, $c=7$
From both the methods we got the same result.