Problem 37 - 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 \(=-\frac{1}{2}\) and \(y\) -intercept \(=4\)

Step-by-Step Solution

Answer

The line with x-intercept of \(-\frac{1}{2}\) and y-intercept of 4 has the following equations: In intercept form: \(8x + y = 4\), In point-slope form: \(y - 4 = \frac{1}{8}x\), and In slope-intercept form: \(y = \frac{1}{8}x + 4\).

Step 1: Calculate the Slope

The slope of the line is calculated using the formula \(-a/b\), where \(a\) and \(b\) are the x and y intercepts respectively. Here, \(a = -\frac{1}{2}\) and \(b = 4\). Thus, the slope, \(m = -(-\frac{1}{2})/4 = 1/8 \)

Step 2: Equation in Intercept Form

The intercept form of the equation of a line is \(x/a + y/b = 1\). Substituting \(a = -\frac{1}{2}\) and \(b = 4\), we get \(-2x + y/4 = 1\), which simplifies to \(8x + y = 4\)

Step 3: Equation in Point-Slope Form

The point-slope form of the equation of a line is \(y - y1 = m(x - x1)\). Here we use the y-intercept point \((0, 4)\) and the slope \(m = 1/8\), so we get \(y - 4 = 1/8 (x - 0)\), which simplifies to \(y - 4 = \frac{1}{8}x\)

Step 4: Equation in Slope-Intercept Form

The slope-intercept form of the equation of a line is \(y = mx + c\), where \(m\) is the slope (which we found to be \(1/8\)) and \(c\) is the y-intercept (which is 4). This simplifies to \(y = \frac{1}{8}x + 4\).