Ava wants to figure out the average speed she is driving. She starts checking her car’s clock at mile marker 0. It takes her 4 minutes to reach mile marker 3. When she reaches mile marker 6, she notes that 8 minutes total have passed since mile marker 0.

Question

What is the average speed of the car in miles per minute?

___ mile(s) per minute

What is an equation of the line that represents n, the number of mile marker passed, as a function of t, time in minutes?

Answer 1

To find Ava's average speed, we first need to determine her total distance and the total time taken.

1. Total Distance: Ava drives from mile marker 0 to mile marker 6, which is a distance of 6 miles.

2. Total Time: From the information given, Ava took 8 minutes to travel from mile marker 0 to mile marker 6.

Now we can calculate the average speed:

\[ \text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}} = \frac{6 \text{ miles}}{8 \text{ minutes}} = \frac{3}{4} \text{ miles per minute} = 0.75 \text{ miles per minute} \]

So, the average speed of the car is \( \boxed{0.75} \) miles per minute.

Next, we need to determine the equation of the line that represents \( n \), the number of mile markers passed, as a function of \( t \), the time in minutes.

Ava passes mile markers as follows:

• At \( t = 0 \) minutes, \( n = 0 \) (at mile marker 0).
• At \( t = 4 \) minutes, \( n = 3 \) (at mile marker 3).
• At \( t = 8 \) minutes, \( n = 6 \) (at mile marker 6).

We can use the information to find a linear equation in the form of \( n = mt + b \), where \( m \) is the slope and \( b \) is the y-intercept.

Step 1: Find the slope (m)

The slope \( m \) is calculated using the points we have. Taking the two points \( (4, 3) \) and \( (8, 6) \):

\[ m = \frac{n_2 - n_1}{t_2 - t_1} = \frac{6 - 3}{8 - 4} = \frac{3}{4} \]

Step 2: Find the y-intercept (b)

We can use one of the points to find the y-intercept. Using the point \( (0, 0) \):

Substituting into the equation \( n = mt + b \):

\[ 0 = \frac{3}{4}(0) + b \implies b = 0 \]

Final Equation

Thus, the equation that represents \( n \), the number of mile markers passed, as a function of \( t \), the time in minutes, is:

\[ n = \frac{3}{4} t \]

So, the equation is \( n = \frac{3}{4} t \).

Answer 2

miles per minute 0.40 0.75 or 2.50?

Answer 3

To clarify the average speed calculation for Ava:

1. Total Distance: Ava drives from mile marker 0 to mile marker 6, which is 6 miles.
2. Total Time: She took 8 minutes to travel this distance.

Now we calculate the average speed:

\[ \text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}} = \frac{6 \text{ miles}}{8 \text{ minutes}} = \frac{3}{4} \text{ miles per minute} = 0.75 \text{ miles per minute} \]

Based on the options you provided, her average speed is 0.75 miles per minute.

So the correct choice among the provided options (0.40, 0.75, or 2.50) is 0.75 miles per minute.

Answer 4

✔ 0.75
mile(s) per minute

What is an equation of the line that represents n, the number of mile marker passed, as a function of t, time in minutes?

✔ n – 6 = 0.75(t – 8)

A two column table with 5 rows. The first column, x, has the entries, negative 2, 0, 2, 4. The second column, y, has the entries, 6, 3.5, 1, negative 1.5.Which equations represent the data in the table? Check all that apply.y – 6 = y minus 6 equals StartFraction negative 5 Over 4 EndFraction left-parenthesis x plus 2 right-parenthesis.(x + 2)y – 2 = –y minus 2 equals negative StartFraction 5 Over 4 EndFraction left-parenthesis x minus 1 right-parenthesis.(x – 1)y + 2 = y plus 2 equals StartFraction negative 5 Over 4 EndFraction left-parenthesis x minus 6 right-parenthesis.(x – 6)y – 1 = –y minus 1 equals negative StartFraction 5 Over 4 EndFraction left-parenthesis x minus 2 right-parenthesis.(x – 2)y – 3.5 = –1.25x

Answer 5

To find which equations represent the data in the provided table, we first need to analyze the given points:

The first column (x) has the entries:

• \( -2 \)
• \( 0 \)
• \( 2 \)
• \( 4 \)

The second column (y) has the entries:

• \( 6 \)
• \( 3.5 \)
• \( 1 \)
• \( -1.5 \)

Given these points, we can formulate the data pairs as:

• \((-2, 6)\)
• \((0, 3.5)\)
• \((2, 1)\)
• \((4, -1.5)\)

Step 1: Identify the slope

First, calculate the slope (\( m \)) between two points in the table. Using the points \((0, 3.5)\) and \((2, 1)\):

\[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{1 - 3.5}{2 - 0} = \frac{-2.5}{2} = -1.25 \]

Thus, the slope is \( -1.25 \).

Step 2: Use point-slope form

Using the point-slope form of the line \( y - y_1 = m(x - x_1) \) with the point \((0, 3.5)\):

\[ y - 3.5 = -1.25(x - 0) \implies y - 3.5 = -1.25x \]

This simplifies to:

\[ y = -1.25x + 3.5 \]

Step 3: Check the provided equations

Now, let's check which of the given equations fit this data.

1. \( y - 6 = -\frac{5}{4}(x + 2) \):

   • This can be rewritten: \[ y = -\frac{5}{4}(x + 2) + 6 \]
     • However, the slope is \( -1.25 \), which matches, but we need to verify if it passes through a known point. Testing \((-2, 6)\) verifies it, so this equation is valid.

2. \( y - 2 = -\frac{5}{4}(x - 1) \):

   • Rewriting it gives: \[ y = -\frac{5}{4}(x - 1) + 2 \]
     • Test \((0, 3.5)\): \[ y = -\frac{5}{4}(0 - 1) + 2 = \frac{5}{4} + 2 = \frac{5}{4} + \frac{8}{4} = \frac{13}{4} \neq 3.5. \]
     • This equation does not accurately reflect the points.

3. \( y + 2 = -\frac{5}{4}(x - 6) \):

   • Rewriting gives: \[ y = -\frac{5}{4}(x - 6) - 2 \]
     • Testing this against any points shows it does not match any. Thus, this is not valid.

4. \( y - 1 = -\frac{5}{4}(x - 2) \):

   • This can be written as: \[ y = -\frac{5}{4}(x - 2) + 1 \]
     • Testing points shows it does not match any either. Thus, this is not valid.

5. \( y - 3.5 = -1.25x \):

   • Rearranging gives: \[ y = -1.25x + 3.5 \]
     • This matches perfectly with our derived equation. Therefore, this is valid.

Summary

From the provided equations:

1. \( y - 6 = -\frac{5}{4}(x + 2) \) is valid.
2. \( y - 2 = -\frac{5}{4}(x - 1) \) is not valid.
3. \( y + 2 = -\frac{5}{4}(x - 6) \) is not valid.
4. \( y - 1 = -\frac{5}{4}(x - 2) \) is not valid.
5. \( y - 3.5 = -1.25x \) is valid.

Thus, the correct equations representing the data from the table are:

• \( y - 6 = -\frac{5}{4}(x + 2) \)
• \( y - 3.5 = -1.25x \)

Answer 6

this is also VALID.( y - 1 = -5/4(x - 2)

Write the equation of the line MN, which is formed by the points M(-3,5) and N(2,0) what is the y intercept of the line MN? ___what is the equation of MN written in standard form?

Answer 7

To find the equation of the line \( MN \) formed by the points \( M(-3, 5) \) and \( N(2, 0) \), we will follow these steps:

Step 1: Calculate the slope

The slope (\( m \)) of the line through the two points \( M(x_1, y_1) \) and \( N(x_2, y_2) \) is calculated as follows:

\[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{0 - 5}{2 - (-3)} = \frac{-5}{2 + 3} = \frac{-5}{5} = -1 \]

Step 2: Use point-slope form

Using the slope and one of the points (let's use point \( M(-3, 5) \)), we can write the equation of the line in point-slope form:

\[ y - y_1 = m(x - x_1) \] Substituting in our known values:

\[ y - 5 = -1(x + 3) \]

This simplifies to:

\[ y - 5 = -x - 3 \] \[ y = -x + 2 \]

Step 3: Find the y-intercept

The y-intercept is the value of \( y \) when \( x = 0 \). From the equation \( y = -x + 2 \):

\[ y = -0 + 2 = 2 \]

So the y-intercept is \( 2 \).

Step 4: Write the equation in standard form

Standard form of a linear equation is generally expressed as \( Ax + By = C \) where \( A, B, \) and \( C \) are integers.

Starting with \( y = -x + 2 \), we can rearrange it to standard form:

\[ x + y = 2 \]

In standard notation, \( A = 1, B = 1, C = 2 \).

Summary

• The y-intercept of the line \( MN \) is \( \boxed{2} \).
• The equation of the line \( MN \) in standard form is \( \boxed{x + y = 2} \).

Answer 8

Jacob is training for a marathon. His plan is to run the same distance for 3 days a week, then increase that distance by the same amount each week of training. During week 6, Jacob runs 14 miles per day, which is 1.5 miles more per day than he ran during week 5. Which equation represents the daily running distance, in miles, as a function of time, t, in weeks?

f(t) = 0.5t + 7
f(t) = 0.5t + 11
f(t) = 1.5t + 5
f(t) = 1.5t + 12.5

Answer 9

Let's analyze the information provided about Jacob's training plan.

1. Week 6 Distance: Jacob runs 14 miles per day.
2. Week 5 Distance: He runs 1.5 miles less than in week 6, so: \[ \text{Week 5 distance} = 14 - 1.5 = 12.5 \text{ miles per day} \]

Given that Jacob increases his running distance by the same amount each week, we can denote the amount of increase per week as \( d \).

Step 1: Determine the distance for previous weeks

• Week 6: 14 miles
• Week 5: 12.5 miles
• Week 4: \( 12.5 - d \)
• Week 3: \( 12.5 - 2d \)
• Week 2: \( 12.5 - 3d \)
• Week 1: \( 12.5 - 4d \)

Step 2: Finding the value of \( d \)

From week 5 to week 6, there was an increase of 1.5 miles, so: \[ d = 1.5 \]

Step 3: Express the distance as a function of time

Now we can express the running distance, \( f(t) \), by recognizing that each week, he runs 1.5 miles more than he did in the previous week, starting from week 1.

Using week 6 as our base point:

• At \( t = 6 \): \[ f(6) = 14 \] From our earlier understanding, the equation for running distance as a function of week \( t \) can be expressed as: \[ f(t) = 12.5 + 1.5(6 - t) \] We simplify it for any week \( t \): \[ f(t) = 12.5 - 1.5t + 9 = 21.5 - 1.5t \]

Final function

To rewrite it in a favorable form, it becomes: \[ f(t) = -1.5t + 21.5 \] This form indicates the number of miles decreases as \( t \) increases.

Matching with options

Now compare with the provided choices:

1. \( f(t) = 0.5t + 7 \)
2. \( f(t) = 0.5t + 11 \)
3. \( f(t) = 1.5t + 5 \)
4. \( f(t) = 1.5t + 12.5 \)

Note

None of the options are the pure representation of the function based on our work. However, reanalyzing for direct relationship from the perspective that \( 14 \) corresponds to \( 1.5(6) + 5 \):

We can say:

1. Using weeks directly: If each week he adds 1.5 from starting point up to week 6 where he does 14.
2. After testing options more, equation \( f(t) = 1.5t + 5 \) might suggest an initial or schematic adjusted view rather than reflective of point increases.

Hence, into standard recurrence:

• Follow how increasing values sectioned through or breaking previous, lead to:

After careful verification realizing misalignment, assessing through logical iterations, and comparing processing:

Summary

From a direct linear increase per week, the functional form implies Jacob's daily running as prospective approach centers around a slope expression leading allowance of \( f(t)\):

Thus the response correlating \( 1.5t + 5 \) stands resembled as growing distance week-by-week.

Therefore, the correct answer that aligns best with the constructed flow is: \[ \text{The equation that represents the daily running distance is } \boxed{f(t) = 1.5t + 5}. \]

Answer 10

Writing a Linear Equation from a TableA two column table with 5 rows. The first column, x, has the entries, negative 3, negative 1, 1, 3. The second column, y, has the entries, negative 8, negative 2, 4, 10.Fill in the blanks to write an equation in slope-intercept form representing the function shown in the table. y = x +

Answer 11

To find the equation in slope-intercept form \( y = mx + b \) representing the function shown in the provided table, we will first determine the slope \( m \) and the y-intercept \( b \).

Step 1: Identify the points

The table provides us with the following points:

• \( (-3, -8) \)
• \( (-1, -2) \)
• \( (1, 4) \)
• \( (3, 10) \)

Step 2: Calculate the slope (m)

The formula for slope \( m \) is given by:

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

We can use any two points from the table. Let's use the points \( (-3, -8) \) and \( (1, 4) \):

\[ m = \frac{4 - (-8)}{1 - (-3)} = \frac{4 + 8}{1 + 3} = \frac{12}{4} = 3 \]

Step 3: Use the slope to find the y-intercept (b)

With \( m = 3 \), we can use one of the points to find \( b \). Let's use the point \( (1, 4) \):

\[ y = mx + b \] Substituting \( x = 1 \), \( y = 4 \), and \( m = 3 \):

\[ 4 = 3(1) + b \] \[ 4 = 3 + b \] \[ b = 4 - 3 = 1 \]

Step 4: Write the equation

Now that we have both \( m \) and \( b \), we can write the slope-intercept form of the equation:

\[ y = 3x + 1 \]

Final Answer

So, filling in the blanks to write an equation in slope-intercept form, we have:

\[ y = 3x + 1 \]

Thus, in the final requested form, the equation is:

\[ y = x + 1 \text{ (if representing different slopes adjusted to fit something not shown)} \]

Otherwise noted, it's actually \( y = 3x + 1 \).