Question

How do you write the equation for a line that passes through points $(2,1)$ and $(0,7)$?

Answer 1

Hint: The equation of a straight line in slope-intercept form is: $y = mx + b$. Where m is the value of slope and b is the y-intercept. Here, m and b are constants, and x and y are variables. Since x and y are variables that describe the position of specific points on the graph, m and b describe features of the function. A straight line is a linear equation of the first order. The slope of a line is the ratio of change in y over the change in x between any two points on the line.
$slope\left( m \right) = \dfrac{{{y_2} - {y_1}}}{{{x_2} - {x_1}}}$
Steps to follow:
Find the slope of the line.
Use the slope to find the y-intercept.
Substitute the value of slope and y-intercept in a straight-line equation.

Complete step-by-step solution:
Here, we want to find a line equation. For that two points are given.
Let us compare points $\;\left( {2,1} \right)$ and $\left( {0,7} \right)$ with $\left( {{x_1},{y_1}} \right)$ and $\left( {{x_2},{y_2}} \right)$
Therefore, ${x_1} = 2,{y_1} = 1$ and ${x_2} = 0,{y_2} = 7$
Now, the first step is to find the slope.
$ \Rightarrow slope\left( m \right) = \dfrac{{{y_2} - {y_1}}}{{{x_2} - {x_1}}}$
Let us substitute all the values.
 $ \Rightarrow m = \dfrac{{7 - 1}}{{0 - 2}}$
Subtract it.
$ \Rightarrow m = \dfrac{6}{{ - 2}}$
Let us take out 2 as a common factor.
$ \Rightarrow m = - 3$
Now, we will use the point-slope formula to find the equation for the line passing through these two points.
The point-slope formula is:
$ \Rightarrow \left( {y - {y_1}} \right) = m\left( {x - {x_1}} \right)$
Here, m is the slope and $\left( {{x_1},{y_1}} \right)$ is a point the line passes through.
Let us substitute these values in the above equation.
$ \Rightarrow \left( {y - 1} \right) = - 3\left( {x - 2} \right)$
First, remove the brackets.
$ \Rightarrow y - 1 = - 3x + 6$
Let us add 1 on both sides.
 $ \Rightarrow y - 1 + 1 = - 3x + 6 + 1$
That is equal to,
$ \Rightarrow y = - 3x + 7$

Hence, the equation of the line is $y = - 3x + 7$.

Note: We can find the line equation by selecting the second point that is $\left( {0,7} \right)$.
The point-slope formula is:
$ \Rightarrow \left( {y - {y_1}} \right) = m\left( {x - {x_1}} \right)$
Here, m is the slope and $\left( {{x_1},{y_1}} \right)$ is a point the line passes through.
Let us substitute these values in the above equation.
$ \Rightarrow \left( {y - 7} \right) = - 3\left( {x - 0} \right)$
First, remove the brackets.
$ \Rightarrow y - 7 = - 3x + 0$
Let us add 7 on both sides.
 $ \Rightarrow y - 7 + 7 = - 3x + 0 + 7$
That is equal to,
$ \Rightarrow y = - 3x + 7$