Question

Find the equation of line passing through (9, 3), (2, 5).

Answer 1

Hint: Compute the slope of the line using the formula \[m = \dfrac{{({y_2} - {y_1})}}{{({x_2} - {x_1})}}\]where the line is passing through the points $(x_1, y_1) and (x_2, y_2)$.
Compute the y-intercept by substituting the slope in the equation of the line y = mx + c where c denotes the y-intercept and use one of the given points for the values of x and y.
Having found both the slope and the y-intercept, substitute them in the equation y = mx + c to get the required answer.

Complete step by step solution: We are given the coordinates of two points (9, 3), (2, 5).
We need to find the equation of the line passing through these two points.
We know that only a line can pass through two given points.
The general equation of a line when the slope and y-intercept is known is given by
y = mx + c where m is the slope of the line and c is the y-intercept
A y-intercept is the point where the line intersects the y-axis.
But the slope and the y-intercept are not known to us.
So, let’s find the values of these quantities.
The slope of a line when two points $(x_1, y_1) and (x_2, y_2) $ are known is given by
\[m = \dfrac{{({y_2} - {y_1})}}{{({x_2} - {x_1})}}\]
Let’s plug in the values in the above formula to find the slope.
Thus, we have \[m = \dfrac{{({y_2} - {y_1})}}{{({x_2} - {x_1})}} = \dfrac{{(5 - 3)}}{{(2 - 9)}} = \dfrac{2}{{ - 7}} = \dfrac{{ - 2}}{7}\]
Now, we need the y-intercept of the line.
We can use the slope to find the y-intercept of the line by substituting the value of m in the equation y = mx + c. Thus, we have
$ y = mx + c \\
   \Rightarrow y = \dfrac{{ - 2}}{7}x + c \\ $
Now, choose one of the given points to substitute for x and y.
Let (x, y) = (2, 5)
Then we have
\[ y = \dfrac{{ - 2}}{7}x + c \\
\Rightarrow 5 = \dfrac{{ - 2}}{7} \times 2 + c \\
\Rightarrow 5 = \dfrac{{ - 4}}{7} + c \\
\Rightarrow c = 5 + \dfrac{4}{7} = \dfrac{{39}}{7} \\ \]
Thus, using the slope and the y-intercept, the equation of the line passing through (9, 3), (2, 5) is
\[y = \dfrac{{ - 2}}{7}x + \dfrac{{39}}{7}\]
We can remove the denominator by multiplying 7 on both the sides of the equation
7y = -2x + 39
Taking the variables on the left hand side and the constants on the right hand side of the equation we get
2x + 7y = 39.
Hence the required equation is \[y = \dfrac{{ - 2}}{7}x + \dfrac{{39}}{7}\] or 2x + 7y = 39

Note: Now, here’s a shorter way of finding the required equation.
The equation of a line when two points $ (x_1, y_1) and (x_2, y_2) $ are known can be given by
\[\dfrac{{(y - {y_1})}}{{(x - {x_1})}} = \dfrac{{({y_2} - {y_1})}}{{({x_2} - {x_1})}}\]
Let $(x_1, y_1)$ = (9, 3) and (x2, y2) = (2, 5), then we get
\[ \dfrac{{(y - 3)}}{{(x - 9)}} = \dfrac{{(5 - 3)}}{{(2 - 9)}} = \dfrac{2}{{ - 7}} = \dfrac{{ - 2}}{7} \\
\Rightarrow 7(y - 3) = - 2(x - 9) \\
\Rightarrow 7y - 21 = - 2x + 18 \\
\Rightarrow 2x + 7y = 18 + 21 = 39 \\
\Rightarrow 2x + 7y = 39 \\ \]
Hence 2x + 7y = 39 is the required equation of the line.