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To determine the number of inches per foot from the equation \(4y = 48x\), we can solve for \(y\) when \(x = 1\).
Start with the equation: \[ 4y = 48x \]
To find \(y\) when \(x = 1\), substitute \(1\) for \(x\): \[ 4y = 48 \cdot 1 \] \[ 4y = 48 \]
Now, solve for \(y\): \[ y = \frac{48}{4} \] \[ y = 12 \]
This means there are 12 inches per foot.
To derive the equation of a line in the form \(y = mx\) that goes through the origin (0,0), we need to determine the slope \(m\) of the line. This slope represents the change in \(y\) for a unit change in \(x\).
To find the slope \(m\), you need two points on the line. For instance, if the graph provides points (x1, y1) and (x2, y2), then the slope \(m\) is calculated using the formula:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Once you identify the slope \(m\), you can write the equation of the line:
\[ y = mx \]
Without the specific image or graph to reference, please follow the steps above with the points on the graph to find the slope and derive the equation. If you can provide the coordinates of two points from the graph, I can help you calculate the slope and formulate the equation.
Since I currently don't have access to the image and can't see the graph you are referring to, I can guide you on how to determine the number of calories in 1 ounce of soda based on a proportional relationship.
Identify Points on the Graph: Look for points on the graph where the x-axis (ounces of soda) and y-axis (calories) intersect. For example, if the graph shows that at 4 ounces, there are 160 calories, this means:
\[ 4 \text{ ounces} = 160 \text{ calories} \]
Calculate the Calories per Ounce: To find the number of calories per ounce, you would divide the total calories by the number of ounces:
\[ \text{Calories per ounce} = \frac{\text{Total Calories}}{\text{Total Ounces}} = \frac{160 \text{ calories}}{4 \text{ ounces}} = 40 \text{ calories/ounce} \]
Repeat for Other Points: If necessary, check another point on the graph to ensure it's consistent. If the graph shows another point, such as 8 ounces for 320 calories, verify:
\[ \text{Calories per ounce} = \frac{320 \text{ calories}}{8 \text{ ounces}} = 40 \text{ calories/ounce} \]
From these calculations, if the values you determine follow this example, you would conclude that there are 40 calories in 1 ounce of soda.
If you can provide specific points from the graph, I can assist you further!
Without the specific graphs showing the earnings for Employee 1 and Employee 2, I can provide a general approach to determine which employee earns more per hour based on the information that would typically be displayed in such graphs.
Identify Points on the Graphs: Look at the y-axis (which typically represents earnings, like dollars) and the x-axis (which usually represents hours worked).
Calculate the Slope (Earnings Per Hour):
For Employee 1, find two points on the line (for example, at 1 hour and at 4 hours). Let's say it shows earnings of $15 after 1 hour and $60 after 4 hours.
For Employee 2, do the same with their points. Let's say this employee earns $20 after 1 hour and $80 after 4 hours.
Compare the Earnings Per Hour:
If you provide specific details from the graphs (like earnings at specific hours for both employees), I can help you determine the exact hourly wages and make a comparison.
To find the slope of a line using similar triangles, you typically need to look at a right triangle formed by the line on a graph. The slope (\(m\)) of a line is defined as the rise over the run, or how much the line goes up (rise) compared to how much it goes across (run).
Here’s how to find the slope using similar triangles:
Identify Two Points on the Line: Look for two clear points on the line, which you can call (x₁, y₁) and (x₂, y₂).
Determine the Rise and Run:
Calculate the Slope:
Use Similar Triangles: If you draw additional lines parallel to the axes from the points to form a right triangle with the x and y axes, you can use similar triangles to verify the slope. The triangles formed by the line will be similar, meaning their corresponding sides are in the same ratio.
Example:
In this example, the slope of the line would be 1.
If you can describe the specific points or provide the graph data, I can help you calculate the specific slope directly. Otherwise, follow the above method by identifying points on your graph.
To write the equation of a line that is parallel to the given line \(y = -3x - 2\) but does not share any points, we need to ensure that the new line has the same slope but a different y-intercept.
Identify the slope of the original line: The slope (m) of the line \(y = -3x - 2\) is \(-3\).
Form the equation of the parallel line: Since parallel lines have the same slope, the equation of a parallel line can be written in the form: \[ y = -3x + b \] where \(b\) (the y-intercept) must be different from \(-2\) to ensure the lines do not intersect.
Choose a different y-intercept: Any value for \(b\) other than \(-2\) will work.
From your responses, the options provided include:
The first option \(y = -3x + 8\) has the same slope of \(-3\) but a different y-intercept (\(8\) instead of \(-2\)). Therefore, this is a valid equation for a parallel line.
The appropriate choice is: \(y = -3x + 8\).
This line is parallel to the original line and does not share any points.
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