Question

How do you write an equation of a line given point $\left( {0,1} \right)$ and point $\left( {5,3} \right)$ ?

Answer 1

Hint: In order to write the equation of a line, we first need to find the slope of the line using the given points, using the formula we have found the value of the slope, we place it in the formula to find the equation of the given line.

Formula used: $slope{\text{ m}} = \dfrac{{{y_2} - {y_1}}}{{{x_2} - {x_1}}}$,
Equation of a line: $y - {y_1} = m\left( {x - {x_1}} \right)$

Complete step-by-step solution:
The given points are: $\left( {0,1} \right)$ and $\left( {5,3} \right)$. We need to find the equation of a line which passes through the following points. We represent these points as so:
$\left( {{x_1},{y_1}} \right) = \left( {0,1} \right)$
 $\left( {{x_2},{y_2}} \right) = \left( {5,3} \right)$
First, we need to find the slope of the given line. Slope of any given line is represented by $m$
We know that $slope{\text{ m}} = \dfrac{{{y_2} - {y_1}}}{{{x_2} - {x_1}}}$
Placing the values of the variable in the above formula we get:
$m = \dfrac{{3 - 1}}{{5 - 0}} = \dfrac{2}{5}$
Now that we have found the value of the slope, we place this value in the formula to find the required equation of the line formed by the two given points:
The formula to find the equation of the line is: $y - {y_1} = m\left( {x - {x_1}} \right)$ where $y,x$ are constants while ${y_1},{x_1}$ are variables.
$y - {y_1} = m\left( {x - {x_1}} \right)$
Placing the value of the variables, we get:
$ \Rightarrow y - 1 = \dfrac{2}{5}\left( {x - 0} \right)$
Solving it further, we get:
$ \Rightarrow y - 1 = \dfrac{2}{5}x$
Now we add $ + 1$ to both sides of the equation:
$ \Rightarrow y = \dfrac{2}{5}x + 1$, now this is in the standard form of $y = mx + c$

Thus, our required equation of a line formed by the given points is $y = \dfrac{2}{5}x + 1$.

Note: The equation of straight line is mostly represented by the formula: $y = mx + c$
Where $m$ is the slope or the angle that the line makes with the x axis, and $c$ the intercept made on the y-axis.
Some common properties of such lines are:
A line which passes through the origin makes zero intercept on the axes.
A horizontal line has no x-intercept and a vertical line has no y - intercept