Problem 47
Question
For the following exercises, express the equations in slope intercept form (rounding each number to the thousandths place). Enter this into a graphing calculator as \(Y 1,\) then adjust the ymin and ymax values for your window to include where the y-intercept occurs. State your ymin and ymax values. $$ 0.537 x-2.19 y=100 $$
Step-by-Step Solution
Verified Answer
The equation in slope-intercept form is \( y \approx 0.245x - 45.662 \) with ymin = -50 and ymax = 50.
1Step 1: Understand the Slope-Intercept Form
The slope-intercept form of a linear equation is given by \( y = mx + b \), where \( m \) is the slope of the line, and \( b \) is the y-intercept.
2Step 2: Rearrange the Equation
The given equation is \( 0.537x - 2.19y = 100 \). We need to solve for \( y \) to rearrange this into slope-intercept form.
3Step 3: Isolate the y-term
Start by adding \( 2.19y \) on both sides. The equation becomes \( 0.537x = 2.19y + 100 \). Then, subtract 100 to get \( 0.537x - 100 = 2.19y \).
4Step 4: Solve for y
Divide every term by 2.19 to isolate \( y \) on one side: \[y = \frac{0.537}{2.19}x - \frac{100}{2.19}\]Calculating these values gives: \[y \approx 0.245x - 45.662\]
5Step 5: Enter into Graphing Calculator
Input \( Y1 = 0.245x - 45.662 \) into the graphing calculator.
6Step 6: Adjust the y-axis Window
Since the y-intercept is \(-45.662\), set \( ymin \) to \(-50\) and \( ymax \) to include positive values, such as \( 50 \), for better viewing of the graph.
7Step 7: Confirm the Window Settings
Ensure that the y-intercept is visible within the window settings, allowing for a clear view of where the line crosses the y-axis.
Key Concepts
Linear EquationsGraphing CalculatorY-InterceptSlope
Linear Equations
Linear equations are equations of the first degree, meaning they have no exponents greater than one. These equations represent straight lines when graphed on a coordinate plane. The standard form of a linear equation is often written as \( Ax + By = C \), where \( A \), \( B \), and \( C \) are constants. The goal with such equations is to express them in different forms that can reveal more information about the line, like its slope and intercepts.
One common form is the slope-intercept form, written as \( y = mx + b \). Here, \( m \) represents the slope, and \( b \) is the y-intercept. This form is particularly useful because it shows how the line behaves as \( x \) changes. By understanding these basic components, you can easily graph linear equations and interpret their meaning.
One common form is the slope-intercept form, written as \( y = mx + b \). Here, \( m \) represents the slope, and \( b \) is the y-intercept. This form is particularly useful because it shows how the line behaves as \( x \) changes. By understanding these basic components, you can easily graph linear equations and interpret their meaning.
Graphing Calculator
A graphing calculator is a powerful tool that can be used to plot equations and visualize their graphs. When working with linear equations, entering them into a graphing calculator can help you see the line and understand how changes in the equation affect its graph.
To enter an equation like the one converted into slope-intercept form \( y = 0.245x - 45.662 \), you should:
To enter an equation like the one converted into slope-intercept form \( y = 0.245x - 45.662 \), you should:
- Access the graphing function on your calculator.
- Input the equation as \( Y1 \).
- Adjust the viewing window to ensure the important parts of the graph are visible.
Y-Intercept
The y-intercept of a line is the point where the line crosses the y-axis. In the slope-intercept form of a linear equation \( y = mx + b \), the y-intercept is represented by \( b \). It is the value of \( y \) when \( x = 0 \).
In our example, the equation \( y = 0.245x - 45.662 \) has a y-intercept of \(-45.662\). This means when you graph the line, it will cross the y-axis at the point \( (0, -45.662) \). The y-intercept is crucial for setting up the graph's initial point on the y-axis and helps to establish the overall position of the line on the graph.
Understanding the y-intercept assists not only in graphing but also in analyzing what the graph represents in real-life situations, as it often signifies an initial value or starting point before any changes in \( x \) occur.
In our example, the equation \( y = 0.245x - 45.662 \) has a y-intercept of \(-45.662\). This means when you graph the line, it will cross the y-axis at the point \( (0, -45.662) \). The y-intercept is crucial for setting up the graph's initial point on the y-axis and helps to establish the overall position of the line on the graph.
Understanding the y-intercept assists not only in graphing but also in analyzing what the graph represents in real-life situations, as it often signifies an initial value or starting point before any changes in \( x \) occur.
Slope
The slope of a line describes its steepness or incline. In the slope-intercept form \( y = mx + b \), \( m \) stands for the slope. Mathematically, it is defined as the change in \( y \) divided by the change in \( x \) between two distinct points on the line, which is often expressed as "rise over run."
For the equation \( y = 0.245x - 45.662 \), the slope is \( 0.245 \). This indicates that for every one unit increase in \( x \), \( y \) increases by 0.245 units. The slope can tell you how fast or slow a line is rising or falling as you move from left to right across the graph.
For the equation \( y = 0.245x - 45.662 \), the slope is \( 0.245 \). This indicates that for every one unit increase in \( x \), \( y \) increases by 0.245 units. The slope can tell you how fast or slow a line is rising or falling as you move from left to right across the graph.
- A positive slope means the line rises, moving from left to right.
- A negative slope means the line falls.
- A zero slope indicates a horizontal line.
- An undefined slope corresponds to a vertical line.
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Problem 47
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