Calculate the gradient (slope) m of the line.

Question

ScanSolve AI Tutor · Accepted Answer

Okay, bazuunelson, let's re-solve question 14 for you.

14. (a) Find the equation of a line passing through $(2,3)$ and $(4,7)$

Calculate the gradient (slope) $m$ of the line.
The formula for the gradient is $m = \frac{y_2 - y_1}{x_2 - x_1}$ .
Using points $(x_1, y_1) = (2,3)$ and $(x_2, y_2) = (4,7)$ :
$m = \frac{7-3}{4-2} = \frac{4}{2} = 2$
Use the point-slope form of the equation of a line, $y - y_1 = m(x - x_1)$ .
Using the point $(2,3)$ and $m=2$ :
$y - 3 = 2(x - 2)$
Simplify the equation to the slope-intercept form $y=mx+c$ .
$y - 3 = 2x - 4$ $y = 2x - 4 + 3$ $y = 2x - 1$
The equation of the line is $\boxed{y = 2x - 1}$ .

14. (b) Find the midpoint of the line
Recall the midpoint formula.
For two points $(x_1, y_1)$ and $(x_2, y_2)$ , the midpoint $M$ is given by $M = \left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right)$ .
Substitute the given points $(2,3)$ and $(4,7)$ into the formula.
$M = \left(\frac{2+4}{2}, \frac{3+7}{2}\right)$
Calculate the coordinates of the midpoint.
$M = \left(\frac{6}{2}, \frac{10}{2}\right)$ $M = (3,5)$
The midpoint of the line is $\boxed{(3,5)}$ .

Step-by-step solution