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Chapter 1: Problem 36

Using a Point and Slope, use the point on the line and the slope \(m\) of the line to find three additional points through which the line passes. (There are many correct answers.) $$(3,-2), \quad m=0$$

Short Answer

Expert verified
The three additional points are (-1,-2), (0,-2) and (4,-2).

Step by step solution

01

Identifying the Given Inputs

Identify the given point and slope. Here, the given point is (3,-2) and the slope \(m\) is 0.
02

Applying the Point-Slope Formula

Substitute the point and slope values into the point-slope form of a line equation, which is \(y - y_1 = m(x - x_1)\). Plugging the values we get \(y + 2 = 0(x - 3)\). Simplifying this equation gives us \(y = -2\).
03

Finding Additional Points

Use the simplified line equation \(y = -2\) to find three more points. It can be quickly seen that as the equation is \(y = -2\), the y value for all points on the line will be the same as the y-coordinate of the given point. Hence you can choose any x-coordinate (different from 3) and the corresponding y-coordinate will be -2. Let's choose x-coordinates as -1, 0, and 4, corresponding y-coordinates will be -2. Hence, the three points are: (-1,-2), (0,-2) and (4,-2).

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Key Concepts

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

Understanding Linear Equations
Linear equations are fundamental in algebra and appear in various forms. A linear equation describes a straight line and can be written in several formats, such as slope-intercept form, standard form, or point-slope form. Here, we mainly focus on the point-slope form.

The general point-slope form is:
  • \[ y - y_1 = m(x - x_1) \]
where \( m \) is the slope of the line, and \((x_1, y_1)\) is a point on the line.
This form is particularly useful when you know the slope of the line and a point through which the line passes.
Losing the complexity found in other forms, it provides a clear picture of the line's direction and location.
To solve for \( y \), you rearrange the formula after plugging in your known values for \( m, x_1, \) and \( y_1 \). This allows you to derive an equation which describes every point along the line.
Exploring the Slope of a Line
The slope of a line, often represented by the letter \( m \), is a measure of its steepness. It tells us how quickly a line climbs or falls as it moves from left to right.
  • If the slope is positive, the line rises over time.
  • If the slope is negative, the line falls.
  • A slope of zero means the line is perfectly horizontal.
  • An undefined slope indicates a vertical line.
In the exercise provided, the slope \( m \) is 0, indicating the line is horizontal.
This means for any value of \( x \), the value of \( y \) remains constant.
So, if given a point (like (3, -2) in the problem), this constancy makes it easy to determine other points on the line by picking different \( x \)-values.
Graphing Lines with Point-Slope Form
Graphing lines using the point-slope form of a linear equation is straightforward once you understand the components involved.
Start by plotting the known point on a graph. This gives you a tangible place to anchor your line.
Since the slope is given, you know how to draw the line from the point based on its steepness.
  • For a slope of 0, the process is very simple: draw a horizontal line through the point.
Once the line is plotted, finding additional points becomes an easy exercise.
With the equation from the exercise, \( y = -2 \), you realize that any \( x \)-coordinate will do, because \( y \) is always -2.
Using this strategy, additional points could be plotted for clarity, such as (-1,-2), (0,-2), and (4,-2), establishing the line's path visually.

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Most popular questions from this chapter

Sales The following are the slopes of lines representing annual sales \(y\) in terms of time \(x\) in years. Use the slopes to interpret any change in annual sales for a one-year increase in time. $$\begin{array}{l}{\text { (a) The line has a slope of } m=135 \text { . }} \\\ {\text { (b) The line has a slope of } m=0 \text { . }} \\ {\text { (c) The line has a slope of } m=-40}\end{array}$$

Cost, Revenue, and Profit A roofing contractor purchases a shingle delivery truck with a shingle elevator for \(\$ 42,000\) . The vehicle requires an average expenditure of \(\$ 9.50\) per hour for fuel and maintenance, and the operator is paid \(\$ 11.50\) per hour. (a) Write a linear equation giving the total cost \(C\) of operating this equipment for \(t\) hours. (Include the purchase cost of the equipment.) (b) Assuming that customers are charged \(\$ 45\) per hour of machine use, write an equation for the revenue \(R\) derived from \(t\) hours of use. (c) Use the formula for profit \(P=R-C\) to write an equation for the profit derived from \(t\) hours of use. (d) Use the result of part (c) to find the break-even point- -that is, the number of hours this equipment must be used to yield a profit of 0 dollars.

Intercept Form of the Equation of a line, use the intercept form to find the equation of the line with the given intercepts. The intercept form of the equation of a line with intercepts \((a, 0)\) and \((0, b)\) is $$\frac{x}{a}+\frac{y}{b}=1, a \neq 0, b \neq 0$$ $$\begin{array}{l}{x \text { -intercept: }\left(\frac{2}{3}, 0\right)} \\ {y \text { -intercept: }(0,-2)}\end{array} $$

From the top of a mountain road, a surveyor takes several horizontal measurements \(x\) and several vertical measurements \(y\) as shown in the table \((x\) and \(y\) are measured in feet). $$\begin{array}{|c|c|c|c|c|}\hline x & {300} & {600} & {900} & {1200} \\\ \hline y & {-25} & {-50} & {-75} & {-100} \\ \hline\end{array}$$ $$\begin{array}{|c|c|c|c|}\hline x & {1500} & {1800} & {2100} \\ \hline y & {-125} & {-150} & {-175} \\ \hline\end{array}$$ $$\begin{array}{l}{\text { (a) Sketch a scatter plot of the data. }} \\\ {\text { (b) Use a straightedge to sketch the line that you }} \\ {\text { think best fits the data. }} \\ {\text { (c) Find an equation for the line you sketched in }} \\ {\text { part (b). }}\end{array}$$ $$ \begin{array}{l}{\text { (d) Interpret the meaning of the slope of the line in }} \\ {\text { part (c) in the context of the problem. }} \\ {\text { (e) The surveyor needs to put up a road sign that }} \\ {\text { indicates the steepness of the road. For instance, }} \\ {\text { a surveyor would put up a sign that states "8% }}\end{array}$$ $$ \begin{array}{l}{\text { grade" on a road with a downhill grade that has }} \\\ {\text { a slope of }-\frac{8 .}{100 .} \text { . What should the sign state for }} \\ {\text { the road in this problem? }}\end{array}$$

Parallel and Perpendicular Lines, determine whether the lines \(L_{1}\) and \(L_{2}\) passing through the pairs of points are parallel, perpendicular, or neither. $$\begin{array}{l}{L_{1} :(-2,-1),(1,5)} \\ {L_{2} :(1,3),(5,-5)}\end{array}$$
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