Problem 31 - 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 \((-3,-2)\) and \((3,6)\)

Step-by-Step Solution

Answer

The point-slope form equation is \(y + 2 = 4/3(x + 3)\) and the slope-intercept form is \(y = 4/3x + 2\).

Step 1: Calculate the slope of the line

To do this, use the formula \(m = (y2 - y1) / (x2 - x1)\). Inserting the given points (-3,-2) and (3,6), we obtain \(m = (6 - (-2)) / (3 - (-3)) = 8/6 = 4/3\). Therefore, the slope of the line, m, is 4/3.

Step 2: Write the equation in Point-Slope Form

To find the equation in point-slope form, use the formula \(y - y1 = m(x - x1)\). In our case, we can use the point (-3,-2), and we know that m=4/3, so the equation becomes \(y - (-2) = 4/3 * (x - (-3))\). Simplifying this, we have \(y + 2 = 4/3(x + 3)\). This is the equation in point-slope form.

Step 3: Convert to Slope-Intercept Form

To rewrite the equation in slope-intercept form, you need to isolate 'y' on one side of the equation, meaning we will have an equation in the form \(y = mx + b\). Rewriting our equation we get: \(y + 2 = 4/3x + 4\). Further simplifying, we get \(y = 4/3x + 2\). This is the equation in slope-intercept form.