Problem 38 - College Algebra Essentials

Question

Use the given conditions to write an equation for each line in point-slope form and slope-intercept form. \(x\) -intercept \(=4\) and \(y\) -intercept \(=-2\)

Step-by-Step Solution

Answer

The equivalent point-slope form is \(y = 1/2x-2\) and the slope-intercept form is \(y = 1/2x - 2\).

Step 1 - Calculation of the slope

Firstly, the slope of the line needs to be calculated using the formula \(m = (y_2 - y_1) / (x_2 - x_1)\). Here, \((x_1, y_1) = (4, 0)\) and \((x_2, y_2) = (0, -2)\). So, \(m = (-2 - 0) / (0 - 4) = 1/2.\)

Step 2 - Construction of equation in point-slope form

The point-slope form of an equation can be written as \((y - y_1) = m(x - x_1)\). Substituting the slope \(m = 1/2\) and point \((4, 0)\), the equation becomes: \((y - 0) = 1/2(x - 4)\) or \(y = 1/2x-2.\)

Step 3 - Construction of equation in slope-intercept form

The slope-intercept form of an equation is given by \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. Substituting for \(m = 1/2\) and \(b = -2\) yields the equation: \(y = 1/2x - 2.\)