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Chapter 1: Problem 1

Find the equation of the line in the form \(y=m x+b\) : (a) through (1,1) and (-5,-3) . (b) through (-1,2) with slope -2 . (c) through (-1,1) and (5,-3) . (d) through (2,5) and parallel to the line \(3 x+9 y+6=0\). (e) with \(x\) -intercept 5 and perpendicular to the line \(y=2 x+4\).

Short Answer

Expert verified
(a) \(y = \frac{2}{3}x + \frac{1}{3}\); (b) \(y = -2x\); (c) \(y = -\frac{2}{3}x + \frac{1}{3}\); (d) \(y = -\frac{1}{3}x + \frac{17}{3}\); (e) \(y = -\frac{1}{2}x + \frac{5}{2}\).

Step by step solution

01

Find the slope for Line (a)

To find the slope of the line passing through the points (1,1) and (-5,-3), use the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]Substitute the given points:\[ m = \frac{-3 - 1}{-5 - 1} = \frac{-4}{-6} = \frac{2}{3} \]
02

Write the equation for Line (a)

With the slope \(m = \frac{2}{3}\) and point (1,1), use the point-slope form: \[ y - y_1 = m(x - x_1) \]Substitute the values:\[ y - 1 = \frac{2}{3}(x - 1) \]Simplify to get the slope-intercept form:\[ y = \frac{2}{3}x + \frac{1}{3} \]
03

Write the equation for Line (b)

The line passes through (-1,2) and has a slope of -2. Use the point-slope form:\[ y - y_1 = m(x - x_1) \]Substitute the values:\[ y - 2 = -2(x + 1) \]Simplify:\[ y = -2x \]
04

Find the slope for Line (c)

Find the slope of the line through (-1,1) and (5,-3) using:\[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-3 - 1}{5 + 1} = \frac{-4}{6} = -\frac{2}{3} \]
05

Write the equation for Line (c)

Using \(m = -\frac{2}{3}\) and point (-1,1):\[ y - 1 = -\frac{2}{3}(x + 1) \]Simplify:\[ y = -\frac{2}{3}x + \frac{1}{3} \]
06

Find the slope for Line (d)

The line is parallel to \(3x + 9y + 6 = 0\). First, express this line in slope-intercept form:\[ 9y = -3x - 6 \Rightarrow y = -\frac{1}{3}x - \frac{2}{3} \]The slope is \(-\frac{1}{3}\), so it's the same for the parallel line.
07

Write the equation for Line (d)

The line passes through (2,5) with the slope found in Step 6:\[ y - 5 = -\frac{1}{3}(x - 2) \]Simplify:\[ y = -\frac{1}{3}x + \frac{17}{3} \]
08

Find the slope for Line (e)

A line perpendicular to \(y = 2x + 4\) has a slope that is the negative reciprocal of 2, which is -\(\frac{1}{2}\).
09

Write the equation for Line (e)

The line has an x-intercept of 5, meaning it passes through (5,0):\[ y - 0 = -\frac{1}{2}(x - 5) \]Simplify:\[ y = -\frac{1}{2}x + \frac{5}{2} \]

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Key Concepts

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

Slope
The concept of the slope is crucial when discussing lines. The slope of a line quantifies its steepness and direction. It is calculated as the "rise over run," or the change in the y-values divided by the change in the x-values between two points on the line. For example, to find the slope between the points (1,1) and (-5,-3), we use the formula:
  • \( m = \frac{y_2 - y_1}{x_2 - x_1} \)
  • Substitute the coordinates: \( m = \frac{-3 - 1}{-5 - 1} = \frac{-4}{-6} = \frac{2}{3} \)
The positive slope of \( \frac{2}{3} \) indicates the line is increasing as it moves from left to right.
Point-Slope Form
The point-slope form is a method used to write the equation of a line when you know the slope and a point on the line. The general form of this equation is:
  • \( y - y_1 = m(x - x_1) \)
Where \( m \) is the slope and \( (x_1, y_1) \) is a point on the line. For instance, using the slope from the previous example, \( m = \frac{2}{3} \), and a point on the line, say (1,1), the equation becomes:
  • \( y - 1 = \frac{2}{3}(x - 1) \)
This can be simplified further to obtain an equation in the slope-intercept form.
Slope-Intercept Form
The slope-intercept form is one of the most common ways to express the equation of a line. It is written as:
  • \( y = mx + b \)
Here, \( m \) represents the slope, and \( b \) represents the y-intercept, which is the point where the line crosses the y-axis. For example, from the previous step, the point-slope form:
  • \( y - 1 = \frac{2}{3}(x - 1) \)
can be rearranged to:
  • \( y = \frac{2}{3}x + \frac{1}{3} \)
This form easily shows both the slope \( \frac{2}{3} \) and the y-intercept \( \frac{1}{3} \).
Parallel Lines
Parallel lines run equidistant from each other and never intersect. In algebra, parallel lines have the same slope. To determine the equation of a line that is parallel to another, simply use the slope of the given line. For example, consider the line given by the equation:
  • \( 3x + 9y + 6 = 0 \)
First, convert this to the slope-intercept form:
  • \( y = -\frac{1}{3}x - \frac{2}{3} \)
The slope is \( -\frac{1}{3} \), so any line parallel to this must also have a slope of \( -\frac{1}{3} \). Then, using a point through which the parallel line passes, such as (2,5), the point-slope form helps find the desired equation.
Perpendicular Lines
Perpendicular lines intersect at a right angle (90 degrees). The slopes of two perpendicular lines are negative reciprocals of each other. If a line has a slope of \( m \), then a line perpendicular to it will have a slope of \( -\frac{1}{m} \). For instance, consider the line:
  • \( y = 2x + 4 \)
This line has a slope of 2; a line that is perpendicular will thus have a slope of \( -\frac{1}{2} \). When you additionally know the line has an x-intercept of 5 (passing through the point (5,0)), the point-slope form gives the equation of this perpendicular line. From here, simplify to find the slope-intercept form easily.

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