Problem 8 - 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 \((-4,2)\) and perpendicular to the line whose equation is \(y=\frac{1}{3} x+7\)

Step-by-Step Solution

Answer

The point-slope form of the line is \(y - 2 = -3(x + 4)\), and the slope-intercept form is \(y = -3x - 10\).

Step 1: Find the slope of the new line

The slope of the line given by the equation \(y=\frac{1}{3}x+7\) is \(\frac{1}{3}\). A line is perpendicular to another if its slope is the negative reciprocal of the other line's slope. Therefore, the slope of the new line is \(-3\).

Step 2: Write the equation of the line in point-slope form

The point-slope form of a line is given by: \(y - y_1 = m(x - x_1)\) where \(m\) is the slope and \((x_1,y_1)\) is a point on the line. Substituting \(-3\) for the slope and \((-4,2)\) for \((x_1,y_1)\), gives: \(y - 2 = -3(x + 4)\) which simplifies to: \(y = -3x - 10\).

Step 3: Write the equation of the line in slope-intercept form

The slope-intercept form of a line is given by \(y = mx + b\) where \(m\) is the slope and \(b\) is the y-intercept. The y-intercept is the y-coordinate of the point where the line crosses the y-axis. From previous steps we know that the slope \(m\) of new line is \(-3\). Plugging this and the given point (-4,2) into the slope-intercept form, we can solve for \(b\): \(2 = -3(-4) + b\). Solving for \(b\) we get \(b = -10\). Thus, the slope-intercept form of the given line is \(y = -3x - 10\).