The graph shows the proportional relationship. Derive the equation of the line y = mx through the origin. (1 point) Π

Question

GPT-4o mini · Answer

In the context of a proportional relationship represented by a graph, the line passes through the origin (0,0), and the equation of this line can be expressed in the form $ y = mx $, where:

$ y $ is the dependent variable,
$ x $ is the independent variable,
$ m $ is the slope of the line.

To derive the equation $ y = mx $, we need to determine the slope $ m $. The slope can be calculated using two points on the line, $ (x_1, y_1) $ and $ (x_2, y_2) $, with the formula:

\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]

Once you determine the value of $ m $, you will have the complete equation of the line in the form $ y = mx $.

If specific points from the graph were provided, you could plug those values into the formula to find $ m $. If you have such points, please share them for a more precise calculation!

Scary spider 🕷️ · Answer

The points on the graph are (1, 5) and (2, 10)

GPT-4o mini · Answer

To derive the equation of the line $ y = mx $ that goes through the origin using the points (1, 5) and (2, 10), we first need to calculate the slope $ m $ using the two points. The formula for slope $ m $ is given by: $$ m = \frac{y_2 - y_1}{x_2 - x_1} $$ Let’s denote the points as: - $ (x_1, y_1) = (1, 5) $ - $ (x_2, y_2) = (2, 10) $ Now, we can substitute these values into the slope formula: $$ m = \frac{10 - 5}{2 - 1} = \frac{5}{1} = 5 $$ Now we have the slope $ m = 5 $. Since the equation of the line that passes through the origin can be written as: $$ y = mx $$ we substitute the value of $ m $: $$ y = 5x $$ Thus, the equation of the line is: $$ \boxed{y = 5x} $$

Scary spider 🕷️ · Answer

Graph the proportional relationship y = 2x What's the points?

GPT-4o mini · Answer

To graph the proportional relationship given by the equation $ y = 2x $, you can find several points by selecting different values for $ x $ and calculating the corresponding $ y $ values. Here are a few points on the line:

When $ x = 0 $: \[ y = 2(0) = 0 \quad \Rightarrow \quad (0, 0) \]
When $ x = 1 $: \[ y = 2(1) = 2 \quad \Rightarrow \quad (1, 2) \]
When $ x = 2 $: \[ y = 2(2) = 4 \quad \Rightarrow \quad (2, 4) \]
When $ x = 3 $: \[ y = 2(3) = 6 \quad \Rightarrow \quad (3, 6) \]
When $ x = -1 $: \[ y = 2(-1) = -2 \quad \Rightarrow \quad (-1, -2) \]
When $ x = -2 $: \[ y = 2(-2) = -4 \quad \Rightarrow \quad (-2, -4) \]

Points to Plot

$ (0, 0) $
$ (1, 2) $
$ (2, 4) $
$ (3, 6) $
$ (-1, -2) $
$ (-2, -4) $

Graphing

To graph $ y = 2x $:

Plot the points listed above on a coordinate plane.
Draw a straight line through these points, extending the line to the edges of the graph in both directions.

The line will rise steeply through the origin, showcasing a direct proportional relationship, meaning as $ x $ increases, $ y $ increases at a consistent rate (specifically, $ y $ is always twice the value of $ x $).

Scary spider 🕷️ · Answer

The proportional relationship between calories and ounces of ounces is shown in the graph. How many calories are in one ounce? The plotted points (3, 36) , (6, 72) , (9, 108) , (12, 144)

GPT-4o mini · Answer

To determine how many calories are in one ounce from the given points in the proportional relationship, we can analyze the coordinates provided and find the relationship between ounces (represented by $ x $) and calories (represented by $ y $).

The provided points are:

$ (3, 36) $
$ (6, 72) $
$ (9, 108) $
$ (12, 144) $

These points indicate that as the number of ounces increases, the calories also increase proportionally.

To find the calories per ounce, we can look at the change in calories for each ounce:

From $ (3, 36) $:
- When $ x = 3 $ ounces, $ y = 36 $ calories.
- Therefore, calories per ounce = $ \frac{36 \text{ calories}}{3 \text{ ounces}} = 12 $ calories/ounce.
Checking another point $ (6, 72) $:
- When $ x = 6 $ ounces, $ y = 72 $ calories.
- Therefore, calories per ounce = $ \frac{72 \text{ calories}}{6 \text{ ounces}} = 12 $ calories/ounce.
Checking the point $ (9, 108) $:
- When $ x = 9 $ ounces, $ y = 108 $ calories.
- Therefore, calories per ounce = $ \frac{108 \text{ calories}}{9 \text{ ounces}} = 12 $ calories/ounce.
Checking the point $ (12, 144) $:
- When $ x = 12 $ ounces, $ y = 144 $ calories.
- Therefore, calories per ounce = $ \frac{144 \text{ calories}}{12 \text{ ounces}} = 12 $ calories/ounce.

Conclusion

In all cases, the calculation shows that there are 12 calories in one ounce.

Thus, the answer is: \[ \boxed{12 \text{ calories per ounce}} \]