A graph shows income per hour, you get 60 dollars at 4 hours

Question

What is the constant of proportionality?
Write an equation that represents this graph
.

c. This is a proportional relationship because the y-intercept is
.

Answer 1

To find the constant of proportionality in the context of income per hour and to construct the equation that represents this relationship, let's break it down step by step.

1. Constant of Proportionality: The income per hour can be calculated by finding the income earned for each hour worked. Here, you earn $60 for 4 hours of work.

   To find the income per hour (constant of proportionality, \( k \)): \[ k = \frac{\text{Total Income}}{\text{Total Hours}} = \frac{60}{4} = 15 \] So, the constant of proportionality is $15. This means you earn $15 for every hour worked.

2. Equation Representation: In a proportional relationship, the equation can be represented as: \[ y = kx \] where:

   • \( y \) is the total income,
   • \( x \) is the number of hours worked,
   • \( k \) is the constant of proportionality (in this case, $15).

   Therefore, the equation that represents this graph is: \[ y = 15x \]

3. Proportional Relationship: A proportional relationship can be identified by checking whether it goes through the origin (0,0). This relationship is proportional because there is a constant ratio (in this case, the ratio of income to hours worked is constant).

   Thus, you would conclude:

   • This is a proportional relationship because the y-intercept is 0 (it passes through the origin).

In summary:

• The constant of proportionality is $15.
• The equation is \( y = 15x \).
• This is a proportional relationship because the y-intercept is 0.

Answer 2

A graph shows a negative slope with two dots
Dot 1: (0, 5)
Dot 2: (6, 1)

Answer 3

Given the two points on the graph, Dot 1: (0, 5) and Dot 2: (6, 1), we can determine the slope of the line, the equation of the line, and confirm details about the relationship represented by these points.

1. Finding the Slope

The slope \( m \) of a line through two points \((x_1, y_1)\) and \((x_2, y_2)\) is calculated using the formula:

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

Substituting the coordinates of the given points:

• \((x_1, y_1) = (0, 5)\)
• \((x_2, y_2) = (6, 1)\)

Now, substituting into the slope formula:

\[ m = \frac{1 - 5}{6 - 0} = \frac{-4}{6} = -\frac{2}{3} \]

2. Equation of the Line

Using the point-slope form of a linear equation, which is:

\[ y - y_1 = m(x - x_1) \]

We can use one of the points. Let's use Dot 1 (0, 5):

\[ y - 5 = -\frac{2}{3}(x - 0) \]

This simplifies to:

\[ y - 5 = -\frac{2}{3}x \] \[ y = -\frac{2}{3}x + 5 \]

3. Summary of the Relationship

The equation \( y = -\frac{2}{3}x + 5 \) represents a linear relationship with a negative slope, indicating that as \( x \) increases, \( y \) decreases. The y-intercept is \( 5 \) (the value of \( y \) when \( x = 0 \)), and the point where the line crosses the y-axis is (0,5).

Conclusion

• Slope: \( -\frac{2}{3} \)
• Equation of the line: \( y = -\frac{2}{3}x + 5 \)
• The relationship is linear with a negative slope, decreasing as \( x \) increases. The y-intercept confirms that it starts at \( (0, 5) \).

Answer 4

(1 point)
Responses

y=3x+5y is equal to 3 x plus 5

y=12x−1y is equal to 1 half x minus 1

y=−45x+3y is equal to negative 4 fifths x plus 3

y=5/4x+3

Answer 5

It looks like you've provided a set of equations along with descriptions. Let’s clarify and analyze each equation systematically:

1. Equation: \( y = 3x + 5 \)

   • This is a linear equation with a positive slope of 3 and a y-intercept of 5. It indicates that for every increase of 1 in \( x \), \( y \) increases by 3.

2. Equation: \( y = 12x - 1 \)

   • This is another linear equation with a slope of 12 and a y-intercept of -1. This means that \( y \) increases by 12 for each unit increase in \( x \).

3. Equation: \( y = -45x + 3 \)

   • This equation has a very steep negative slope of -45 and a y-intercept of 3. This indicates that \( y \) decreases dramatically as \( x \) increases.

4. Equation: \( y = \frac{5}{4}x + 3 \)

   • This equation has a positive slope of \( \frac{5}{4} \) and a y-intercept of 3. Each increase of 1 in \( x \) results in an increase of \( \frac{5}{4} \) in \( y \).

Summary of the Characteristics:

• The first and second equations have positive slopes, indicating a direct relationship between \( x \) and \( y \).
• The third equation has a steep negative slope, indicating an inverse relationship between \( x \) and \( y \).
• The fourth equation also has a positive slope, but less steep than the first two.

If you have a specific question or need further explanation related to these equations, please clarify!

Answer 6

A graph shows a negative slope with two dots
Dot 1: (0, 5)
Dot 2: (6, 1)
Which equation represents the graph?

(1 point)
Responses

y=3x+5y is equal to 3 x plus 5

y=12x−1y is equal to 1 half x minus 1

y=−45x+3y is equal to negative 4 fifths x plus 3

y=54x+3

Answer 7

To find which equation represents the graph with points (0, 5) and (6, 1), we first need to determine the correct equation from the provided responses based on the slope and y-intercept of the line defined by those two points.

Step 1: Calculate the Slope

As calculated in a previous message, we find the slope \( m \) using the two points:

• Dot 1: \((0, 5)\)
• Dot 2: \((6, 1)\)

The slope is calculated as follows:

\[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{1 - 5}{6 - 0} = \frac{-4}{6} = -\frac{2}{3} \]

Step 2: Find the Equation of the Line

Using the slope-intercept form \( y = mx + b \):

We can use the y-intercept from Dot 1 (0, 5) to find \( b \):

Since the y-intercept is 5, our equation becomes:

\[ y = -\frac{2}{3}x + 5 \]

Step 3: Evaluating Provided Options

None of the provided equations exactly match the derived equation. However, let us analyze each one:

1. \( y = 3x + 5 \): Positive slope, does not match.
2. \( y = 12x - 1 \): Positive slope, does not match.
3. \( y = -45x + 3 \): Extremely steep negative slope, does not match.
4. \( y = 54x + 3 \): Positive slope, does not match.

Conclusion

Among the options provided, none of the equations accurately represent the graph of the line through the points (0, 5) and (6, 1).

The correct line equation based on the points given should be approximately:

\[ y = -\frac{2}{3}x + 5 \]

If you have further instructions or options, please provide them!

Answer 8

Which equation represents the graph?

(1 point)
Responses

y=3x+5y is equal to 3 x plus 5

y=12x−1y is equal to 1 half x minus 1

y=−45x+3y is equal to negative 4 fifths x plus 3

y=5/4x+3

Answer 9

To determine which equation represents the graph formed by the points (0, 5) and (6, 1), we already established that the slope of the line is \(-\frac{2}{3}\) and the y-intercept is \(5\). Therefore, the expected equation of the line is:

\[ y = -\frac{2}{3}x + 5 \]

Now let's evaluate the provided options to see which one might be a reasonable match:

1. Option 1: \(y = 3x + 5\)

   • This has a positive slope, not matching our line.

2. Option 2: \(y = 12x - 1\)

   • Again, this has a positive slope and does not match.

3. Option 3: \(y = -45x + 3\)

   • This has a steep negative slope, which is far from \(-\frac{2}{3}\).

4. Option 4: \(y = \frac{5}{4}x + 3\)

   • This has a positive slope, not matching.

Conclusion

None of the provided options match the equation \(y = -\frac{2}{3}x + 5\). Therefore, the correct equation representing the graph formed by the points (0, 5) and (6, 1) is not among the options given.

If you need further assistance or more options related to this question, feel free to ask!

Answer 10

Just pick one

Answer 11

Given the options you've provided, I will select Option 3: \( y = -45x + 3 \) as the closest choice because it is the only equation with a negative slope. However, please note that it does not accurately represent the graph through the points (0, 5) and (6, 1), as the slope in the correct equation is \(-\frac{2}{3}\) and not \(-45\).

If you are required to choose one regardless, Option 3 is the best fit from the limited choices.