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Chapter 2: Problem 9

Write an equation for each line in the indicated form. Write the equation in point-slope form for the line that passes through (0,4) and is perpendicular to the line \(x-\) \(2 y=6\).

Short Answer

Expert verified
The equation in point-slope form is \( y - 4 = -2x \).

Step by step solution

01

Convert the Original Equation to Slope-Intercept Form

The given equation is \( x - 2y = 6 \). To find the slope, we first need to rearrange it into slope-intercept form \( y = mx + b \). Begin by solving for \( y \): \(-2y = -x + 6\). Divide every term by \(-2\) to get \( y = \frac{1}{2}x - 3 \). Therefore, the slope of this line is \( \frac{1}{2} \).
02

Find the Slope of the Perpendicular Line

Lines that are perpendicular have slopes that are negative reciprocals of each other. If the slope of the given line is \( \frac{1}{2} \), then the slope of the perpendicular line is the negative reciprocal, which is \(-2\).
03

Use the Point-Slope Formula

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. We are given the point \((0, 4)\) and the perpendicular slope \(-2\). Substitute these values into the formula: \( y - 4 = -2(x - 0) \).
04

Simplify the Equation

Simplify the equation \( y - 4 = -2(x - 0) \). This simplifies to \( y - 4 = -2x \), which is the point-slope form of the equation.

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Key Concepts

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

Perpendicular Lines
Perpendicular lines intersect at a right angle (90 degrees). In the world of coordinate geometry, two lines are perpendicular if the product of their slopes is -1. This means that their slopes are negative reciprocals.
  • If you know the slope of one line, you can find the slope of a line perpendicular to it by taking the negative reciprocal.
  • For example, if one line has a slope of \( \frac{1}{2} \), then a line perpendicular to it will have a slope of \(-2\).
  • This concept is essential when you're trying to determine the relationship between two lines on the coordinate plane.
Understanding how to find perpendicular lines is crucial in various mathematical applications, including geometry and algebra.
Slope-Intercept Form
The slope-intercept form of a line gives you a quick way to write the equation of a line. It's expressed as \( y = mx + b \), where:
  • \( m \) is the slope of the line, representing the steepness or tilt.
  • \( b \) is the y-intercept, which is the point where the line crosses the y-axis.
For the equation given in the original problem, \( x - 2y = 6 \), we rearranged it to slope-intercept form to find the slope. After calculation, we found it to be \( y = \frac{1}{2}x - 3 \), meaning:
  • The slope \( (m) \) is \( \frac{1}{2} \).
  • The y-intercept \( (b) \) is \(-3 \).
This form is particularly useful when you want to graph a line quickly or compare different lines.
Negative Reciprocal
The concept of a negative reciprocal is key when working with perpendicular lines in geometry. To find the negative reciprocal of a number, you follow these steps:
  • First, flip the number (take its reciprocal).
  • Then, change the sign of the flipped number.
In our example from the exercise, with an initial slope of \( \frac{1}{2} \):
  • The reciprocal of \( \frac{1}{2} \) is \( 2 \).
  • The negative reciprocal is \( -2 \), which becomes the slope of the perpendicular line.
This method of determining slopes is powerful for understanding how different lines relate to one another on a graph.
Equation of a Line
Determining the equation of a line is a fundamental skill in algebra and is essential for understanding linear relationships.
  • Equations of lines can be written in various forms, but the point-slope form is commonly used when you know a point on the line and the slope.
In this format, the equation is expressed as \( y - y_1 = m(x - x_1) \):
  • \( m \) is the slope of the line.
  • \( (x_1, y_1) \) is a specific point on the line.
In our case, we used the point \((0, 4)\) and the slope \(-2\) to construct the equation. Substituting these values, we simplified to get:
  • \( y - 4 = -2(x - 0) \)
  • This simplifies further to \( y = -2x + 4 \).
This equation now succinctly describes the line that fulfills both conditions: passing through the given point and being perpendicular to the original line.

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Most popular questions from this chapter

Write each statement in simplified interval notation. $$ x \geq-5 \text { and } x<2 $$

Write an equation for each line in the indicated form. Write the equation in slope-intercept form for the line that passes through (-2,2) and is perpendicular to the line \(x+3 y=8\)

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