Problem 21 - College Algebra Essentials

Question

Use the given conditions to write an equation for each line in point-slope form and slope-intercept form. Slope \(=\frac{1}{2},\) passing through the origin

Step-by-Step Solution

Answer

The point-slope form and slope-intercept form of the given line is \(y = \frac{1}{2}x\).

Step 1: Write the Point-Slope Form of a Line

The point-slope form of a line is given as \(y - y_1= m(x-x_1)\). In this scenario, the slope (\(m\)) is \(\frac{1}{2}\) , and the point through which the line passes (\(x_1, y_1\)) is (0,0), so the equation becomes \(y - 0 = \frac{1}{2}(x - 0)\). Simplifying this equation, we get \(y = \frac{1}{2}x\).

Step 2: Write the Slope-Intercept Form of a Line

The slope-intercept form of a line is usually represented as \(y = mx + b\), where \(m\) is the slope, and \(b\) is the y-intercept. Given the slope (\(m = \frac{1}{2}\)), and knowing that our line passes through the origin, we know that \(b = 0\). Thus, replacing \(m\) and \(b\) in the equation, we get \(y = \frac{1}{2}x + 0\). This simplifies to the same equation: \(y = \frac{1}{2}x\).