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Chapter 1: Problem 14

Find equations for these lines: a. Passes through the points \((-1,3)\) and \((4,1)\). b. \(x\) intercept \((3,0)\) and \(y\) intercept \(\left(0,-\frac{2}{3}\right)\) c. Contains \((-1,3)\) and is perpendicular to \(5 x-3 y=7\)

Short Answer

Expert verified
a. \( y = -\frac{2}{5}x + \frac{16}{5} \) b. \( y = \frac{2}{9}x - \frac{2}{3} \) c. \( y = -\frac{3}{5}x + \frac{12}{5} \)

Step by step solution

01

Find the slope of the line passing through two points

For Part (a), to find the equation of the line passing through points (-1, 3)and(4, 1), we first need to determine the slope. The slope formula is \( m = \frac{y_2 - y_1}{x_2 - x_1} \). Plug in the values: \( m = \frac{1 - 3}{4 + 1} = \frac{-2}{5} \).
02

Use the point-slope form to find the line equation

Now that the slope \( m \)is determined, use the point-slope form of a line equation: \( y - y_1 = m(x - x_1) \). Choose one of the points, let's use (-1, 3)\: \( y - 3 = -\frac{2}{5}(x + 1) \). Rearrange it to get the slope-intercept form: \( y = -\frac{2}{5}x + \frac{1}{5} + 3 \Rightarrow y = -\frac{2}{5}x + \frac{16}{5} \).
03

Determine the slope using intercepts

For Part (b), using the intercepts given (x:3,0) and (y:0, -\frac{2}{3}),\,we can find the slope m \.Slope \( m \)is calculated by the rise over run:\( m = \frac{\Delta y}{\Delta x} = \frac{-\frac{2}{3} - 0}{0 - 3} = \frac{-\frac{2}{3}}{-3} = \frac{2}{9} \).
04

Write the line equation using intercepts form

In part (b), we now use the intercepts form of the equation: \( y = mx + c \). We know the slope,\( m = \frac{2}{9}\), and the y-intercept,\( -\frac{2}{3} \), so substitute these values in: \( y = \frac{2}{9}x - \frac{2}{3} \).
05

Determine perpendicular slope in part (c)

For Part (c), a line perpendicular to a given line has a slope that is the negative reciprocal. The given line鈥檚 equation is \( 5x - 3y = 7 \). Rearrange it to slope-intercept form to find the slope: \( 3y = 5x - 7 \Rightarrow y = \frac{5}{3}x - \frac{7}{3} \). Thus the slope is \( m_1 = \frac{5}{3} \). The slope of the perpendicular line,\( m_2 \), would be the negative reciprocal: \( m_2 = -\frac{3}{5} \).
06

Use the point-slope form for perpendicular line

Now, use the point-slope form of the line equation with the point (-1, 3): \( y - 3 = -\frac{3}{5}(x + 1) \). Rearrange to get the slope-intercept form: \( y - 3 = -\frac{3}{5}x - \frac{3}{5} \Rightarrow y = -\frac{3}{5}x + \frac{12}{5} \).

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Key Concepts

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

slope of a line
The slope of a line is a measure of how steep the line is. To find the slope between two points, use the formula: \( m = \frac{y_2 - y_1}{x_2 - x_1} \).The symbols \( y_1 \) and \( y_2 \) represent the y-coordinates of the two points, while \( x_1 \) and \( x_2 \) are the x-coordinates. The slope tells us the ratio of the vertical change (rise) to the horizontal change (run). For example, if you have two points \((-1,3)\) and \((4,1)\), follow these steps:
  • Subtract the y-coordinates: \(1 - 3 = -2\).
  • Subtract the x-coordinates: \(4 + 1 = 5\).
  • Slope \( m = \frac{-2}{5} \).
A positive slope means the line is increasing, while a negative slope means the line is decreasing.
point-slope form
The point-slope form of a line's equation is useful when you know one point on the line and the slope. It is written as: \( y - y_1 = m(x - x_1) \)where \((x_1, y_1)\) is the known point and \( m \) is the slope. For example, if given the point \((-1, 3)\) and the slope \(\frac{-2}{5}\), you can write: \( y - 3 = \frac{-2}{5}(x + 1) \). This form is great for quickly finding the equation of a line but usually needs to be rearranged to either slope-intercept form or standard form.
slope-intercept form
The slope-intercept form of a line's equation is probably the most common. It looks like this: \( y = mx + c \). Here, \( m \) is again the slope and \( c \) is the y-intercept (the point where the line crosses the y-axis). For instance, using the point-slope form example: \( y - 3 = \frac{-2}{5}(x + 1) \), we can rearrange to get: \( y = \frac{-2}{5}x + \frac{16}{5} \). This tells us that the line crosses the y-axis at \( \frac{16}{5} \) and has a slope of \( \frac{-2}{5} \).
perpendicular lines
Perpendicular lines intersect at a right angle (90 degrees). To find the equation of a line that is perpendicular to another, you need to find the negative reciprocal of the original line's slope. This means if the original line's slope is \( m \), the slope of the perpendicular line will be\( -\frac{1}{m} \). For instance, if we have a line with equation \( 5x - 3y = 7 \), rearrange it to slope-intercept form: \( 3y = 5x - 7 \), or \( y = \frac{5}{3}x - \frac{7}{3} \). So its slope is\( \frac{5}{3} \), and the slope of the perpendicular line will be \( \frac{-3}{5} \). Now pass this line through the point \((-1, 3)\): \( y - 3 = \frac{-3}{5}(x +1) \). Rearrange to get: \( y = \frac{-3}{5}x + \frac{12}{5} \). This is the equation of the line perpendicular to \( 5x - 3y = 7 \).

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Most popular questions from this chapter

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