Problem 29 - College Algebra Essentials

Question

Use the given conditions to write an equation for each line in point-slope form and slope-intercept form. Passing through \((-3,-1)\) and \((2,4)\)

Step-by-Step Solution

Answer

The point-slope form of the line is \(y + 1 = x + 3\), and the slope-intercept form is \(y = x + 2\).

Step 1: Calculate the Slope

Use the given coordinates \((-3,-1)\) and \((2,4)\) to find the slope \(m\). The formula for slope is \(m = \frac{(y_2 - y_1)}{(x_2 - x_1)}\). Plugging the points into this formula gives: \(m = \frac{(4 - (-1))}{(2 - (-3))} = \frac{5}{5} = 1.\)

Step 2: Point-Slope Form

With the slope \(m = 1\) and one pair of coordinates, for instance \((-3,-1)\), the point-slope form of the line can be written using the formula: \(y - y_1 = m(x - x_1)\). Plugging in the known values gives: \(y - (-1) = 1(x - (-3))\), which simplifies to \(y + 1 = x + 3\). This is the point-slope form of the line.

Step 3: Slope-Intercept Form

The slope-intercept form \(y = mx + b\) can be found by solving the point-slope equation for \(y\). This gives \(y = x + 3 - 1\), which simplifies to \(y = x + 2\). This is the slope-intercept form of the line.