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Chapter 3: Problem 233

In the following exercises, find the equation of each line. Write the equation in slope-intercept form. . Perpendicular to the line \(x-2 y=5,\) point (-2,2)

Short Answer

Expert verified
The equation of the line is y = -2x - 2

Step by step solution

01

Find the slope of the given line

First, rewrite the equation of the line in slope-intercept form, which is of the form y = mx + b. Start with the given equation: x - 2y = 5 To solve for y, subtract x from both sides: -2y = -x + 5 Then, divide everything by -2 to isolate y: y = (1/2)x - 5/2 So, the slope (m) of the given line is 1/2.
02

Determine the slope of the perpendicular line

The slope of a line perpendicular to another line is the negative reciprocal of the original line's slope. The slope of the line given is 1/2. The negative reciprocal of 1/2 is -2. Therefore, the slope of the perpendicular line is -2.
03

Use the point-slope form to find the equation of the line

Now that the slope of the perpendicular line is known (-2) and the point it passes through (-2, 2), use the point-slope form of the equation of a line: y - y1 = m(x - x1) Substitute m = -2, x1 = -2, and y1 = 2 into the equation: y - 2 = -2(x + 2)
04

Simplify the equation to slope-intercept form

Simplify the equation to get it into slope-intercept form (y = mx + b): y - 2 = -2(x + 2) y - 2 = -2x - 4 Add 2 to both sides to isolate y: y = -2x - 2

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

slope-intercept form
The slope-intercept form is a way to write the equation of a line. It is written as:

\[y = mx + b\]
In this form, \(m\) represents the slope of the line, and \(b\) is the y-intercept, the point where the line crosses the y-axis. This form is particularly useful because it clearly shows both the slope and the y-intercept, making it easy to graph the line.

For example, if we have the equation \(y = 2x + 3\), we know:
  • slope (\(m\)) is 2
  • y-intercept (\(b\)) is 3
To convert an equation to slope-intercept form, you simply need to solve for \(y\). This often involves isolating \(y\) on one side of the equation.
perpendicular lines
Perpendicular lines are lines that intersect at a right angle (90 degrees). The slopes of perpendicular lines have a special relationship: they are negative reciprocals of each other.

If you have a line with slope \(m\), the slope of a line perpendicular to it will be \(-1/m\).

For example, if one line has a slope of 3, the perpendicular line will have a slope of \(-1/3\).

This inverse relationship helps us find the slope of the perpendicular line quickly. In our exercise, the given line has a slope of \(1/2\). Thus, the slope of the perpendicular line is \(-2\) since \(-2\) is the negative reciprocal of \(1/2\).
point-slope form
Point-slope form is another way to write the equation of a line. It uses the slope of the line and a point that the line passes through. The formula is:

\[y - y1 = m(x - x1)\]
Here, \(m\) is the slope, and \((x1, y1)\) is a point on the line.

This form is particularly helpful when you have a point and a slope but not the y-intercept.

In our exercise, we have a point (-2, 2) and a slope (-2). Plugging these values into the point-slope form formula, we get:

\[y - 2 = -2(x + 2)\]
We use point-slope form to derive the equation of the line, which can then be converted into slope-intercept form for simplicity.
negative reciprocal
The term negative reciprocal is crucial when dealing with perpendicular lines. A reciprocal is simply one divided by the number. For example, the reciprocal of 2 is \(\frac{1}{2}\).

A negative reciprocal changes the sign of the reciprocal. So, the negative reciprocal of 2 is \(-1/2\) and the negative reciprocal of \(1/2\) is \(-2\).

In our exercise, the slope of the given line is \(1/2\). Therefore, the slope of any perpendicular line will be \(-2\), the negative reciprocal of \(1/2\).

Understanding this concept helps to quickly find the slope of a line perpendicular to a given one, which is essential for many geometry problems.

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