Question

Write the point-slope form of the equation of the line that passes through the two points. $$ (-2,3),(2,-5) $$

Short answer

The equation of the line that passes through the points (-2,3) and (2,-5) is \( y = -2x + 7 \) in point-slope form.

Step-by-step solution

Calculate the Slope

The slope of a line passing through the points \((-2,3)\) and \((2,-5)\) is given by the formula \( m = \frac{{y2 - y1}}{{x2 - x1}} \). Substituting the given points into this formula gives: \(m = \frac{{-5 - 3}}{{2 - (-2)}} = \frac{{-8}}{{4}} = -2 \). So, the slope of the line is -2.

Substitute into the Point-Slope Formula

The point-slope form of a line is given by \( y - y1 = m(x - x1) \). Substituting the slope -2 found in Step 1, and one of the points, say (-2,3), into this formula gives: \( y - 3 = -2(x - (-2)) \). Simplifying this gives \( y - 3 = -2x - 4 \).

Rewrite in Standard Form

The equation \( y - 3 = -2x - 4 \) can be rewritten in standard form by first distributing the -2 through the parentheses, which gives \( y - 3 = -2x + 4 \). Then, isolating y in the equation results in \( y = -2x + 7 \).