Problem 26 - 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 \((3,5)\) and \((8,15)\)

Step-by-Step Solution

Answer

The line passing through the points (3, 5) and (8, 15) can be represented by the equation \(y - 5 = 2(x - 3)\) in point-slope form and \(y = 2x - 1\) in slope-intercept form.

Step 1: Calculate the slope

We can calculate the slope of the line using the formula \(m = \frac{y_2-y_1}{x_2-x_1}\). Plugging the given points into this formula gives \(m = \frac{15-5}{8-3} = \frac{10}{5} = 2\).

Step 2: Write the equation in point-slope form

Substituting \(m = 2\), \(x_1 = 3\), and \(y_1 = 5\) into the equation for point-slope form, \(y - y_1 = m(x - x_1)\), we get \(y - 5 = 2(x - 3)\).

Step 3: Convert the equation into slope-intercept form

Solving the equation from Step 2 for \(y\) gives the equation in slope-intercept form. \(y - 5 = 2x - 6 \Rightarrow y = 2x - 1\).