Question

How do you write an equation of a line going through $(0,7),(3,5)$?

Answer 1

Hint: Use the equation of a line using two points formula. Firstly, find the slope by dividing the difference of $y$ coordinates with the difference of $x$-coordinates and then substitute the values of the given coordinates in the formula and evaluate it in order to get a linear equation in two variables.

Formulas used:
The formula for finding slope when two points are given is,$m = \dfrac{{({y_2} - {y_1})}}{{({x_2} - {x_1})}}$
The formula for finding the equation of a line for two points is,
$({y_{}} - {y_1}) = m({x_{}} - {x_1})$

Complete step-by-step answer:
The given coordinates are $(0,7),(3,5)$
Let us give it a notation as,
${x_1} = 0;{y_1} = 7$
${x_2} = 3;{y_2} = 5$
Firstly, we find the slope for the given coordinates.
We use the formula, $m = \dfrac{{({y_2} - {y_1})}}{{({x_2} - {x_1})}}$
On substituting,
$ \Rightarrow m = \dfrac{{(5 - 7)}}{{(3 - 0)}}$
On further evaluation,
$ \Rightarrow m = \dfrac{{ - 2}}{3}$
Now, we use the two point-line formula to get the line equation.
The formula for it is,$(y - {y_1}) = m(x - {x_1})$
Now, substitute the values of $m,{x_1},{y_1}$
$ \Rightarrow (y - 7) = \dfrac{{ - 2}}{3}(x - 0)$
Continue simplifying,
$ \Rightarrow y - 7 = \dfrac{{ - 2(x)}}{3}$
Multiply both the sides of the equation with $3$,
$ \Rightarrow 3(y - 7) = \dfrac{{ - 2x}}{3}(3)$
$ \Rightarrow 3y - 21 = - 2x$
On regrouping the terms,
$ \Rightarrow 2x + 3y = 21$
Representing it in slope-intercept form,
$ \Rightarrow y = \dfrac{{ - 2x}}{3} + 7$
where $7$ is the $y$ intercept.

$\therefore $The line equation formed by two coordinates $(0,7),(3,5)$ is $2x + 3y = 21$.

Additional information: The slope of a line is the steepness of a line in a horizontal or vertical direction. The slope of a line can be calculated by taking the ratio of the change in vertical dimensions upon the change in horizontal dimensions.

Note:
While substituting values in any formula one must check if the values are correctly placed in their respective positions or else it would lead to negative values or incorrect values which has a huge impact on the graphical representation of the equations.