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Chapter 1: Problem 74

Find a linear equation whose graph is the straight line with the given properties. Through \((p, q)\) and \((r, s)(p \neq r)\)

Short Answer

Expert verified
To find a linear equation passing through points \((p, q)\) and \((r, s)\) with \(p \neq r\), first calculate the slope \(m = \frac{s - q}{r - p}\). Then, use the slope-intercept form \(y = mx + b\), and substitute one of the points to find the y-intercept \(b = q - m \cdot p\). Finally, write the equation as \(y = mx + b\) with the calculated values of \(m\) and \(b\).

Step by step solution

01

Calculate the slope of the line

To find the slope (m) of the line passing through two points \((p, q)\) and \((r, s)\), we use the formula: $$m = \frac{s - q}{r - p}$$ Since we know that \(p \neq r\), we won't have any issue computing the slope.
02

Use the slope-intercept form

Once we have the slope of the line, we can use the slope-intercept form of a linear equation, which is: $$y = mx + b$$ Here, "m" is the slope, and "b" is the y-intercept. We already have the slope, so now we just need to find the y-intercept, "b."
03

Substitute one of the points and the slope into the equation

To find the y-intercept, "b," we can use either of the points given, \((p, q)\) or \((r, s)\). For this example, we'll use the point \((p, q)\). Substitute the point's values and the slope into the slope-intercept equation: $$q = m \cdot p + b$$ Now we have an equation with one unknown, "b," which we can solve for.
04

Solve for the y-intercept, "b."

Rearrange the equation from Step 3 and solve for "b": $$b = q - m \cdot p$$ This will give us the y-intercept of the line.
05

Write the final linear equation

Now that we have both the slope, "m," and the y-intercept, "b," we can write the final equation for the linear function: $$y = mx + b$$ Substitute the values for "m" and "b" that we found in the previous steps to get the final equation of the line passing through the points \((p, q)\) and \((r, s)\). That's the linear equation for the line that goes through the given points.

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Key Concepts

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

Slope of a Line
Understanding the slope of a line is crucial when working with linear equations. The slope is a measure of how steep a line is and the direction in which it inclines or declines. To find the slope of a line that passes through two specific points, like \text{(p, q)} and \text{(r, s)}, we use the formula \( m = \frac{s - q}{r - p} \).

This formula gives us the ratio of the vertical change (rise) to the horizontal change (run) between the two points. The slope can tell us a lot about the line; for instance, if the slope is positive, the line rises from left to right. Conversely, a negative slope indicates that the line falls from left to right. If the slope is zero, the line is horizontal, indicating no rise or fall. Understanding the slope helps to visualize and predict the behavior of the line over its entire length.
Slope-Intercept Form
The slope-intercept form of a linear equation is one of the most commonly used formats to showcase the properties of a line. It is expressed as \( y = mx + b \), where \( m \) represents the slope of the line, and \( b \) denotes the y-intercept, the value of y where the line crosses the y-axis.

In many cases, this form is practical because it directly gives you the slope and y-intercept, making it easier to graph the line or understand how it behaves with just a glance. When given the slope and a point through which the line passes, one can easily determine the y-intercept by rearranging terms and solving for \( b \). This form also makes it straightforward to compare two lines. For instance, if they have the same slope, they are parallel, and if the product of their slopes is negative one, they are perpendicular.
Y-Intercept
The y-intercept is a specific point where a line crosses the y-axis of a coordinate plane. In the slope-intercept form of a linear equation \( y = mx + b \), the y-intercept is denoted by \( b \).

This value represents the y-coordinate of the point at which the line will intersect the y-axis when the value of x is zero. Finding the y-intercept is a key step in graphing linear equations and understanding their properties. It's often the starting point when drawing a line on a graph, as you can plot the y-intercept first and then use the slope to determine the direction and steepness of the line. For instance, if you have a point \( (p, q) \), and the slope \( m \), you can plug these values into the equation and rearrange to find \( b \), which gives you a clear starting point for the line on the graph.

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Most popular questions from this chapter

The total weekly revenue earned at Royal Ruby Retailers is given by $$ R(p)=-\frac{4}{3} p^{2}+80 p $$ where \(p\) is the price (in dollars) RRR charges per ruby. Use this function to determine: a. The weekly revenue, to the nearest dollar, when the price is set at \$20/ruby. b. The weekly revenue, to the nearest dollar, when the price is set at \(\$ 200\) /ruby. (Interpret your result.) c. The price \(\mathrm{RRR}\) should charge in order to obtain a weekly revenue of \(\$ 1,200\).

Suppose that \(y\) is decreasing at a rate of 4 units per 3-unit increase of \(x\). What can we say about the slope of the linear relationship between \(x\) and \(y ?\) What can we say about the intercept?

A piano manufacturer has a daily fixed cost of \(\$ 1,200\) and a marginal cost of \(\$ 1,500\) per piano. Find the cost \(C(x)\) of manufacturing \(x\) pianos in one day. Use your function to answer the following questions: a. On a given day, what is the cost of manufacturing 3 pianos? b. What is the cost of manufacturing the 3rd piano that day? c. What is the cost of manufacturing the 11 th piano that day? d. What is the variable cost? What is the fixed cost? What is the marginal cost? e. Graph \(C\) as a function of \(x\). HINT [See Example 1.]

Compute the sum-of-squares error \((S S E)\) by hand for the given set of data and linear model. $$ (0,-1),(1,3),(4,6),(5,0) ; \quad y=-x+2 $$

In the Fahrenheit temperature scale, water freezes at \(32^{\circ} \mathrm{F}\) and boils at \(212^{\circ} \mathrm{F}\). In the Celsius scale, water freezes at \(0^{\circ} \mathrm{C}\) and boils at \(100^{\circ} \mathrm{C}\). Assuming that the Fahrenheit temperature \(F\) and the Celsius temperature \(C\) are related by a linear equation, find \(F\) in terms of \(C\). Use your equation to find the Fahrenheit temperatures corresponding to \(30^{\circ} \mathrm{C}, 22^{\circ} \mathrm{C},-10^{\circ} \mathrm{C}\), and \(-14^{\circ} \mathrm{C}\), to the nearest degree.
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