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Chapter 3: Problem 64

The equation of line \(A\) is given. Write the equation in slope-intercept form of the line (line \(B\) ) that is perpendicular to line \(A\) and that passes through the given point. $$ y=\frac{3}{4} x+2 ;(6,-15) $$

Short Answer

Expert verified
\( y = -\frac{4}{3}x - 7 \)

Step by step solution

01

- Identify the slope of line A

The given equation of line A is in slope-intercept form (y = mx + b), where m is the slope. Thus, the slope of line A is \( \frac{3}{4} \).
02

- Determine the slope of the perpendicular line B

The slopes of two perpendicular lines are negative reciprocals of each other. Therefore, if the slope of line A is \( \frac{3}{4} \), the slope of line B will be \( -\frac{4}{3} \).
03

- Use the point-slope form equation

Using the slope from Step 2 and the given point (6, -15), substitute these values into the point-slope form of a line equation: \( y - y_1 = m(x - x_1) \). So, \( y + 15 = -\frac{4}{3}(x - 6) \).
04

- Simplify the equation

Expand and simplify the equation from Step 3 to put it in slope-intercept form. First, distribute the slope: \( y + 15 = -\frac{4}{3}x + 8 \). Next, isolate y: \( y = -\frac{4}{3}x + 8 - 15 \), which simplifies to \( y = -\frac{4}{3}x - 7 \).

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Key Concepts

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

Negative Reciprocals
In the context of perpendicular lines, negative reciprocals play a crucial role. When two lines are perpendicular, the slopes of these lines multiply to -1. This unique relationship is represented through negative reciprocals. For a slope of a line \(m\), if you have another line perpendicular to it, the slope of this new line is \(-1/m\). For instance, if a line has a slope of \(\frac{3}{4}\), then a line perpendicular to it will have a slope of \(-\frac{4}{3}\). This relationship ensures the two lines intersect at a right angle.
Point-Slope Form
Point-slope form of a linear equation is especially useful when a point on the line and its slope are known. The point-slope form is written as \(y-y_1=m(x-x_1)\), where \(m\) is the slope and \((x_1, y_1)\) is a specific point the line goes through. For example, let's consider we have a point \( (6, -15)\) and the slope is \(-\frac{4}{3}\). Plugging these into the point-slope form gives \(y + 15 = -\frac{4}{3}(x - 6)\). This formula helps us easily find the required linear equation when combined with slope information.
Perpendicular Lines
Perpendicular lines intersect at a 90-degree angle. They have a distinctive property where their slopes are negative reciprocals. This means if one line has a slope \( m\), the slope of the line perpendicular to it will be \( -1/m\). In our given problem, if line A has a slope of \(\frac{3}{4}\), then the perpendicular line B has a slope of \(-\frac{4}{3}\). This unique relationship is used to determine the equation of one line given the properties of another.
Linear Equations
Linear equations represent straight lines on a graph and are typically written in slope-intercept form \( y = mx + b br> Here, \(m\) denotes slope, and \(b\) is the y-intercept. Linear equations are fundamental in algebra and represent the simplest forms of equations describing linear relationships. In solving the exercise, translating the point-slope form equation into slope-intercept form helps in understanding and graphing the line easily. Starting from our point-slope form \(y + 15 = -\frac{4}{3}(x - 6)\), after simplifying, we get the form of \ y = -\frac{4}{3}x - 7\), representing our final linear equation in slope-intercept form.

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Most popular questions from this chapter

Use the slope formula to find the slope of the line that passes through the points. \(\left(\frac{1}{3}, \frac{4}{5}\right) ;\left(\frac{5}{3}, \frac{2}{5}\right)\)

For exercises 1-8, (a) represent the information as two ordered pairs. (b) find the average rate of change, \(m\). The amount of certified organic cropland in Washington State planted in peas increased from 28 acres in 2007 to 252 acres in 2010. Round to the nearest whole number. (Source: www.tfrec.wsu.edu, March 2011)

(a) graph the given points, and draw a line through the points. (b) use the graph to find the slope of the line. (c) use the slope formula to find the slope of the line. \((-3,-4) ;(1,3)\)

The completed problem has one mistake. (a) Describe the mistake in words, or copy down the whole problem and highlight or circle the mistake. (b) Do the problem correctly. Problem: Find the \(x\)-intercept of \(9 x+2 y=36\). Incorrect Answer: \(9 x+2 y=36\) $$ \begin{aligned} 9(0)+2 y &=36 \\ 2 y &=36 \\ \frac{2 y}{2} &=\frac{36}{2} \\ y &=18 \end{aligned} $$ The \(x\)-intercept is \((0,18)\).

The completed problem has one mistake. (a) Describe the mistake in words, or copy down the whole problem and highlight or circle the mistake. (b) Do the problem correctly. Problem: Use the slope formula to find the slope of the line that passes through \((6,2)\) and \((6,7)\). $$ \text { Incorrect Answer: } \begin{aligned} m &=\frac{y_{2}-y_{1}}{x_{2}-x_{1}} \\ m &=\frac{6-6}{7-2} \\ m &=\frac{0}{5} \\ m &=0 \end{aligned} $$
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