Question

How do you write an equation in slope-intercept form given that the line passes through the point $(2,7)$ and has a slope of $2$?

Answer 1

Let us first be familiar with linear equations in two variables. An equation in the form of $ax + by = c$, where $a$, $b$ and $c$ are constants and $x$ and $y$ are variables, is known as a linear equation in two variables. We must know that the graph of such equations is a straight line. Now, there are many ways to express the graph of a straight line. One such way is the slope-intercept form. It is expressed as $y = mx + c$, where $x$ and $y$ are the coordinates of the point through which the line passes, $m$ is the slope of the line and $c$ is the $y$-intercept of the line.

Complete step by step solution:
Given point through which the line passes is $(2,7)$, such that
$\Rightarrow (x,y) = (2,7)$
$\Rightarrow x = 2$ and $\Rightarrow y = 7$
Also, it is given that the slope of the line is $2$, such that
$\Rightarrow m = 2$
Since there is nothing given about the $y$-intercept of the line, we will have to find it.
We have $x = 2$, $y = 7$ and $m = 2$. On substituting these value in the equation of slope intercept form of the line, we get
$\Rightarrow y = mx + c$
$\Rightarrow 7 = 2 \times 2 + c$
On simplifying and interchanging the left-hand side and the right-hand side of the equations, we get
$\Rightarrow 4 + c = 7$
On taking $4$ to the right-hand side of the equation, we will get
$\Rightarrow c = 7 - 4$
$\Rightarrow c = 3$
Now we have $c = 3$. On substituting the values of $m$ and $c$ in the equation of the slope-intercept form of the line, we get
$\Rightarrow y = mx + c \\\ \Rightarrow y = 2x + 3 \\\$

Hence, when a line passes through the point $(2,7)$ and has a slope of $2$, its equation in the slope-intercept form can be given as $y = 2x + 3$.

Note:
The slope of the line is the measure of incline or steepness of the line. Whereas, the $y$-intercept of the line is the point where it crosses the $y$-axis in the graph.