Calculate the gradient (slope) m of the line.
This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
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14. (a) Find the equation of a line passing through $(2,3)$ and $(4,7)$
Step 1: 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 $(x_2, y_2) = (4,7)$:
$$m = \frac{7-3}{4-2} = \frac{4}{2} = 2$$
Step 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 $m=2$:
$$y - 3 = 2(x - 2)$$
Step 3: 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
Step 1: Recall the midpoint formula.
For two points $(x_1, y_1)$ and $(x_2, y_2)$, the midpoint $M$ is given by $M = \left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right)$.
Step 2: Substitute the given points $(2,3)$ and $(4,7)$ into the formula.
$$M = \left(\frac{2+4}{2}, \frac{3+7}{2}\right)$$
Step 3: Calculate the coordinates of the midpoint.
$$M = \left(\frac{6}{2}, \frac{10}{2}\right)$$
$$M = (3,5)$$
The midpoint of the line is $\boxed{(3,5)}$.
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