Problem 85 - Algebra and Trigonometry

Question

Write an equation in point-slope form and slope-intercept form for the line passing through \((-2,-6)\) and perpendicular to the line whose equation is \(x-3 y+9=0 .\) (Section 2.4 Example \(2)\)

Step-by-Step Solution

Answer

The equation of the line passing through the point (-2,-6) and perpendicular to the given line in point-slope form is \(y + 6 = -3(x + 2)\) and in slope-intercept form is \(y = -3x - 12\).

Step 1: Finding the slope of the original line

The equation of the original line is given by \(x - 3y + 9 = 0\). By rearranging this into 'y' equal format which is \( y = \frac{1}{3}x - 3 \), it can be seen that the slope of this line is \(\frac{1}{3}\).

Step 2: Find the slope of the perpendicular line

The slope of a line perpendicular to the original line would be the negative reciprocal of the original line's slope. So, the slope of the line perpendicular to the given line would be \(-\frac{1}{\frac{1}{3}}\) or -3.

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

The point-slope form of the equation of a line is \(y - y_1 = m(x - x_1)\), where \(m\) is the slope and \((x_1, y_1)\) is a point on the line. Here, \(m = -3\) and the point \((x_1, y_1)\) is \((-2, -6)\), so the equation in point-slope form is \(y - (-6) = -3(x - (-2))\) or \(y + 6 = -3(x + 2)\).

Step 4: Rewrite the equation in slope-intercept form

We can now rewrite the equation we just found in slope-intercept form (which is \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept). By rearranging, we get \(y = -3x - 6 - 6\) or \(y = -3x - 12\).

Key Concepts

Point-Slope Form

When you have a line and you know a point on it, as well as its slope, you can easily write its equation using the point-slope form. This form is given by:\[ y - y_1 = m(x - x_1) \]- Here, \( m \) represents the slope of the line.- \((x_1, y_1)\) is a known point on the line.Given the point (-2, -6) and the slope (-3) (because it's perpendicular to the original line with slope of (\frac{1}{3})), you would plug these values straight into the formula. This results in:\[ y + 6 = -3(x + 2) \]This equation represents our line in the point-slope form. It's quite useful as it directly incorporates both a given point and the slope.

Slope-Intercept Form

The slope-intercept form is another convenient way to express the equation of a line. It is written as:\[ y = mx + b \]- \( m \) is the slope, which tells you how steep the line is.- \( b \) is the y-intercept, the point where the line crosses the y-axis.From the point-slope form equation:\[ y + 6 = -3(x + 2) \]By expanding and rearranging, we convert it to slope-intercept form:1. Distribute \(-3\) to both \(x\) and \(2\): \[ y + 6 = -3x - 6 \]2. Subtract \(6\) from both sides: \[ y = -3x - 12 \]Now we have the line's equation in slope-intercept form. It shows that the line crosses the y-axis at (-12) and has a slope of (-3).

Negative Reciprocal

Understanding negative reciprocals is key to finding perpendicular lines. The negative reciprocal of a slope is used to find the slope of any line perpendicular to another.- A reciprocal flips the fraction; for instance, the reciprocal of \(\frac{1}{3}\) is \(3\).- Adding a negative changes its sign, so the negative reciprocal of \(\frac{1}{3}\) is (-3).Negative reciprocals are important because the product of the slopes of two perpendicular lines is always (-1). In this exercise, the original line has a slope of (\frac{1}{3}), and thus its perpendicular line has a slope of (-3). This shows how quickly using reciprocals helps simplify understanding line relationships.