Problem 104
Question
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. $$\frac{x}{\frac{1}{2}}+\frac{y}{5}=1$$
Step-by-Step Solution
Verified Answer
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 \(y = 5\).
1Step 1: 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.
2Step 2: 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).
3Step 3: 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.
4Step 4: 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.
Key Concepts
Slope-Intercept FormX-InterceptY-InterceptGraphing Utility
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:
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.
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.
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 \( y = -2x + 5 \).
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.
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.
Other exercises in this chapter
Problem 103
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
View solution Problem 104
(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, a
View solution Problem 105
Use the functions \(f(x)=\frac{1}{8} x-3\) and \(g(x)=x^{3}\) to find the indicated value or function. $$\left(f^{-1} \circ g^{-1}\right)(1)$$
View solution Problem 105
Use the results of Exercises 101–104 to write an equation of the line that passes through the points. \(x\) -intercept: (2,0) \(y\) -intercept: (0,9)
View solution