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There are 3 racers on a graph racer one starts at point (0,8),(4,9),(8,10)

y =
Racer two
Points (0,6),(2,7(,(4,8)(6,9),(8,10)
y =

5 answers

To find the equations of the lines representing the two racers based on the points you've provided, we'll first determine the slopes and then the equations of the lines using the slope-intercept form \(y = mx + b\) where \(m\) is the slope and \(b\) is the y-intercept.

Racer One:

The points given are (0, 8), (4, 9), and (8, 10).

To find the slope \(m\) between two points, we can use the formula:

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

Let's find the slope using the first two points (0, 8) and (4, 9):

  • \(x_1 = 0\), \(y_1 = 8\)
  • \(x_2 = 4\), \(y_2 = 9\)

\[ m = \frac{9 - 8}{4 - 0} = \frac{1}{4} \]

Now using the slope and one of the points (0, 8) to determine \(b\):

Using the equation \(y = mx + b\): \[ 8 = \frac{1}{4}(0) + b \implies b = 8 \]

So the equation for Racer One is: \[ y = \frac{1}{4}x + 8 \]

Racer Two:

The points given are (0, 6), (2, 7), (4, 8), (6, 9), and (8, 10).

Using the first two points (0, 6) and (2, 7) to find the slope:

\[ m = \frac{7 - 6}{2 - 0} = \frac{1}{2} \]

Now to determine \(b\) using the point (0, 6): \[ 6 = \frac{1}{2}(0) + b \implies b = 6 \]

So the equation for Racer Two is: \[ y = \frac{1}{2}x + 6 \]

Summary:

  • Racer One: \(y = \frac{1}{4}x + 8\)
  • Racer Two: \(y = \frac{1}{2}x + 6\)
They tell me it’s wrong on both

Let’s go through the calculations step by step for both racers to ensure that we obtain the correct equations for the lines based on the points provided.

Racer One

The points for Racer One are: (0, 8), (4, 9), and (8, 10).

  1. Calculate the slope between the points (0, 8) and (4, 9): \[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{9 - 8}{4 - 0} = \frac{1}{4} \]

  2. Calculate the slope between the points (4, 9) and (8, 10): \[ m = \frac{10 - 9}{8 - 4} = \frac{1}{4} \]

  3. Both calculations give us a consistent slope of \(\frac{1}{4}\).

  4. Using point (0, 8) to find the y-intercept (b): \[ y = mx + b \implies 8 = \frac{1}{4}(0) + b \implies b = 8 \]

The equation becomes: \[ y = \frac{1}{4}x + 8 \]

Racer Two

The points for Racer Two are: (0, 6), (2, 7), (4, 8), (6, 9), and (8, 10).

  1. Calculate the slope between the points (0, 6) and (2, 7): \[ m = \frac{7 - 6}{2 - 0} = \frac{1}{2} \]

  2. Calculate the slope between the points (2, 7) and (4, 8): \[ m = \frac{8 - 7}{4 - 2} = \frac{1}{2} \]

  3. Calculate the slope between the points (4, 8) and (6, 9): \[ m = \frac{9 - 8}{6 - 4} = \frac{1}{2} \]

  4. Calculate the slope between the points (6, 9) and (8, 10): \[ m = \frac{10 - 9}{8 - 6} = \frac{1}{2} \]

  5. All calculations consistently give us a slope of \(\frac{1}{2}\).

  6. Using point (0, 6) to find the y-intercept (b): \[ y = mx + b \implies 6 = \frac{1}{2}(0) + b \implies b = 6 \]

The equation becomes: \[ y = \frac{1}{2}x + 6 \]

Final Equations

  1. Racer One: \(y = \frac{1}{4}x + 8\)
  2. Racer Two: \(y = \frac{1}{2}x + 6\)

If these equations are marked as incorrect, please verify the points you've provided or let me know what specific details might need adjustment!

Thanks points are accurate but when I put answer in tells me wrong

Let's double-check the data you've provided and work through the equations step by step to ensure everything is correct.

Racer One

Given Points: (0, 8), (4, 9), (8, 10)

  1. Finding the slope (m) using the first two points:

    • Points: (0, 8) and (4, 9) \[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{9 - 8}{4 - 0} = \frac{1}{4} \]

  2. Using the slope with the point (0, 8) to determine the y-intercept (b): \[ y = mx + b \implies 8 = \frac{1}{4}(0) + b \implies b = 8 \]

  3. Equation of Racer One: \[ y = \frac{1}{4}x + 8 \] (This is correct according to the calculations from the given points.)

Racer Two

Given Points: (0, 6), (2, 7), (4, 8), (6, 9), (8, 10)

  1. Finding the slope (m) using the first two points:

    • Points: (0, 6) and (2, 7) \[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{7 - 6}{2 - 0} = \frac{1}{2} \]

  2. Using the slope with the point (0, 6) to determine the y-intercept (b): \[ y = mx + b \implies 6 = \frac{1}{2}(0) + b \implies b = 6 \]

  3. Equation of Racer Two: \[ y = \frac{1}{2}x + 6 \] (This equation is also derived correctly based on the points.)

Re-checking the Equations

  • Racer One: \(y = \frac{1}{4}x + 8\)
  • Racer Two: \(y = \frac{1}{2}x + 6\)

Potential Issues

  1. Input Format: Check how you are entering the equations into your system. Ensure you match the required format exactly. Some systems may require the fraction to be expressed differently or particular notation for explicit functions.

  2. Check for Standard Form: Sometimes, systems require the equation in a specific form, such as:

    • Standard form: \(Ax + By + C = 0\)
    • If required, convert them:
      • Racer One: \(-\frac{1}{4}x + y - 8 = 0\) or \(x - 4y + 32 = 0\) (multiplying everything by 4)
      • Racer Two: \(-\frac{1}{2}x + y - 6 = 0\) or \(x - 2y + 12 = 0\) (multiplying everything by 2)

  3. Verifying Software/Calculator: Ensure that the tool or software you are using does not have specific settings or bugs that may interpret various forms of the equation differently.

If you can share how you are inputting the answer or in what context (homework, software, etc.), I might be able to provide more tailored advice!