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Chapter 1: Problem 104

Use a graphing utility to graph the equation of the line in the form $$\frac{x}{a}+\frac{y}{b}=1, \quad a \neq 0, b \neq 0$$ Use the graphs to make a conjecture about what \(a\) and \(b\) represent. Verify your conjecture. $$\frac{x}{\frac{1}{2}}+\frac{y}{5}=1$$

Short Answer

Expert verified
In the equation \(\frac{x}{a} + \frac{y}{b} = 1\), 'a' is the x-intercept and 'b' is the y-intercept. This was verified by graphing the equation \(\frac{x}{0.5} + \frac{y}{5} = 1\), which showed that the line indeed intersected the x-axis at \(x = 0.5\) and the y-axis at \(y = 5\).

Step by step solution

01

Rearrange the Equation into Slope-Intercept Form

The equation given in the form \(\frac{x}{a} + \frac{y}{b} = 1\) can be rearranged into the typical slope-intercept form \(y = mx + b\), where \(m\) is the slope of the line and \(b\) is the y-intercept. In this case, rearranging the equation into the typical form, we would have \(y = -\frac{b}{a}x + b\). From this transformation, it can be observed that 'a' influences the slope and 'b' dictates the y-intercept of the line.
02

Graph the Equation

Using a graphing utility, plot the equation \(\frac{x}{0.5} + \frac{y}{5} = 1\). This will correspond to the equation \(y = -2x + 5\). This line will be a decreasing straight line with a slope of -2 that intersects y-axis at the point (0,5).
03

Make the Conjecture

Based on the observations so far, we can make a conjecture that in the equation \(\frac{x}{a} + \frac{y}{b} = 1\), 'a' is the x-intercept and 'b' is the y-intercept of the line.
04

Verify the Conjecture

If we place 'a' = 0.5 and 'b' = 5 into the conjecture, we find that when \(x = 0.5\), \(y = 0\) and when \( y = 5 \), \(x = 0 \), confirming the intercept form of the line equation, thus our conjecture is correct.

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Key Concepts

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

Slope-Intercept Form
One of the most essential concepts in graphing linear equations is the slope-intercept form. This form is represented as \( y = mx + b \), where \( m \) indicates the slope of the line, and \( b \) is the y-intercept.
Understanding this form allows us to easily identify two key aspects of the line:
  • The slope \( m \) which describes how steep the line is. A positive slope climbs upwards, while a negative slope descends.
  • The y-intercept \( b \), which tells us where the line crosses the y-axis.
By rearranging equations into this form, we can quickly glean information about the line's direction and positioning on the graph. For the given equation, rearranging gives \( y = -2x + 5 \), showcasing a slope of -2 and a y-intercept at 5.
X-Intercept
The x-intercept of a line is the point where the line crosses the x-axis. At this point, the y-coordinate is zero, so we can find the x-intercept by setting \( y = 0 \) in the line equation and solving for \( x \).
In the equation \( \frac{x}{a} + \frac{y}{b} = 1 \), the value \( a \) directly denotes the x-intercept. This is because substituting \( y = 0 \) will solve to \( x = a \).
For our equation \( \frac{x}{0.5} + \frac{y}{5} = 1 \), setting \( y = 0 \) gives us \( x = 0.5 \). Thus, the x-intercept is 0.5, confirming that point where the line crosses the x-axis.
Y-Intercept
The y-intercept is another crucial component in understanding linear equations. It is where the line meets the y-axis, and is found by setting \( x = 0 \) in the equation.
The equation \( y = mx + b \) makes this easy to find, as the y-intercept is simply \( b \). But looking at the alternative equation \( \frac{x}{a} + \frac{y}{b} = 1 \), the y-intercept \( b \) becomes clear when \( x = 0 \), leading directly to \( y = b \).
For our specific equation, substituting \( x = 0 \) leads to \( y = 5 \), so the y-intercept is 5. This directly aligns with the y-intercept in the slope-intercept form equation \( y = -2x + 5 \).
Graphing Utility
A graphing utility is a tool that helps visualize equations by plotting them on a coordinate plane. This can include graphing calculators or software that can handle equations and inequalities.
Using a graphing utility to plot the line described by \( \frac{x}{0.5} + \frac{y}{5} = 1 \) helps provide a visual confirmation of mathematical concepts such as slope, x-intercept, and y-intercept.
  • Graphing utilities can quickly draw complicated graphs with precision.
  • They are helpful in verifying solutions, testing multiple scenarios, and understanding transformations.
For this exercise, employing a graphing utility confirmed our conversion to the slope-intercept form, allowing us to observe the line intersecting the y-axis at 5 and the x-axis at 0.5, enhancing our comprehension of the conjecture made.

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Most popular questions from this chapter

Use the functions \(f(x)=x+4\) and \(g(x)=2 x-5\) to find the specified function. $$(g \circ f)^{-1}$$

Identify the terms. Then identify the coefficients of the variable terms of the expression. $$\frac{x}{3}-5 x^{2}+x^{3}$$

Let \(y=g(x)\) represent the function that gives the women's European shoe size in terms of \(x,\) the women's U.S. size. A women's U.S. size 6 shoe corresponds to a European size \(37 .\) Find \(g^{-1}(g(6)).\)

(a) use a graphing utility to graph the function \(f,\) (b) use the draw inverse feature of the graphing utility to draw the inverse relation of the function, and (c) determine whether the inverse relation is an inverse function. Explain your reasoning. $$f(x)=\frac{4 x}{\sqrt{x^{2}+15}}$$

Determine whether the equation represents \(y\) as a function of \(x .\) $$x=5$$
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