Problem 64 - College Algebra

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

Step 1: Find the slope of the given line

The equation of the given line is in the form \(y = mx + b\), where \(m\) is the slope of the line. From the equation \(y = \frac {1}{3}x + 7\), one can see that the slope of the given line is \(\frac{1}{3}\).

Step 2: Calculate the slope of the new line

Because the new line is perpendicular to the given line and the slopes of perpendicular lines are negative reciprocals, the new slope will be the negative reciprocal of \(\frac{1}{3}\). This results in a new slope of \( -3\).

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

The point-slope form of a line is \(y - y_1 = m(x - x_1)\) where \((x_1, y_1)\) is a point on the line and \(m\) is the slope of the line. Using the calculated slope of -3 and the given point \((-4, 2)\), the required line in point-slope form is \(y - 2 = -3(x + 4)\).

Step 4: Convert to slope-intercept form

By normal algebraic manipulation, re-arrange \( y - 2 = -3(x + 4) \) into the form \( y = mx + b \), which will be the slope-intercept form of the line. Simplify the equation to get \(y = -3x - 12 + 2\), therefore the final slope-intercept form equation is \(y = -3x - 10\).