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Chapter 3: Problem 44

Determine whether each statement "makes sense" or "does not make sense" and explain your reasoning. In many examples, I use the slope-intercept form of a line's equation to obtain an equivalent equation in point-slope form.

Short Answer

Expert verified
The statement 'In many examples, I use the slope-intercept form of a line's equation to obtain an equivalent equation in point-slope form' makes sense as you can transform a line's equation from one form to the other given a slope and a point on the line.

Step by step solution

01

Understanding the forms

First, understand what the slope-intercept and point-slope forms of a line are. The slope-intercept form is \(y = mx + c\), and the point-slope form is \(y - y_1 = m(x - x_1)\). We can convert from one to another using a known point on the line and the slope.
02

Analyzing the statement

Now let's analyze the statement. It says that 'In many examples, I use the slope-intercept form of a line to obtain an equivalent equation in point-slope form'. This makes sense as you can transform the equation from slope-intercept form to point-slope form given the slope and a point on the line.
03

Example of transformation

Let's take an example. Suppose we have a line in slope-intercept form \(y = 3x + 2\). Now, suppose that we know one of the points on the line is \((2,8)\). We can use this point and the slope to convert it into point-slope form: \(y - 8 = 3(x - 2)\). This illustrates the statement made, proving it makes sense.

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Most popular questions from this chapter

Solve for \(y\) and put the equation in slope-intercept form. $$y-3=4(x+1)$$

Find the coefficients that must be placed in each shaded area so that the equation's graph will be a line with the specified intercepts. \(\square x+\square y=10 ; x\) -intercept \(=5 ; y\) -intercept \(=2\)

a. Graph each of the following points: $$ \left(1, \frac{1}{2}\right),(2,1),\left(3, \frac{3}{2}\right),(4,2) $$ Parts (b)-(d) can be answered by changing the sign of one or both coordinates of the points in part (a). b. What must be done to the coordinates so that the resulting graph is a mirror-image reflection about the \(y\) -axis of your graph in part (a)? c. What must be done to the coordinates so that the resulting graph is a mirror-image reflection about the \(x\) -axis of your graph in part (a)? d. What must be done to the coordinates so that the resulting graph is a straight-line extension of your graph in part (a)?

Use a graphing utility to graph in a standard viewing rectangle, \([-10,10,1]\) by \([-10,10,1]\). Then use the \([\text { TRACE }]\) feature to trace along the line and find the coordinates of two points. $$y=2 x-1$$

graph each linear equation in two variables. Find at least five solutions in your table of values for each equation. $$y=-\frac{5}{2} x+1$$
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