how do you create a two variable linear equation?

Question

GPT-4o mini · Answer

Creating a two-variable linear equation involves identifying a linear relationship between two variables, usually referred to as $ x $ and $ y $. Here's how you can create one step-by-step:

Understand the Format: A two-variable linear equation can be expressed in the standard form as: \[ Ax + By = C \] or in slope-intercept form: \[ y = mx + b \] where:
- $ A $, $ B $, and $ C $ are constants.
- $ m $ is the slope of the line.
- $ b $ is the y-intercept (the value of $ y $ when $ x = 0 $).
Define Your Variables: Decide what your two variables will represent. For example, let $ x $ represent the number of hours studied, and $ y $ represent the score on an exam.
Determine a Relationship: Establish a linear relationship between the two variables. For instance, you might say that for every hour studied, the exam score increases by 10 points, starting from a base score of 50 when no hours are studied.

From this assumption, you can derive the equation.
Write Your Equation:
- If the increase in score is 10 points per hour studied, you can express this relationship as: \[ y = 10x + 50 \]
- In standard form, this would be: \[ -10x + y = 50 \quad \text{or rearranging gives } 10x - y = -50 \]
Verify the Equation: Ensure that the equation represents the intended relationship:
- If $ x = 0 $ (no hours studied), then $ y = 50 $ (base score).
- If $ x = 2 $ (two hours studied), then $ y = 10(2) + 50 = 70 $, which indicates a 20-point increase, consistent with our original relationship.

Now you have a two-variable linear equation. You can also create different equations by adjusting the slope and y-intercept according to different scenarios or relationships.

Answer

A local theater is planning to host a play. The theater has 300 seats available, and it sells two types of tickets: standard tickets (S) and premium tickets (P). The theater wants to find a combination of ticket sales that exactly fills the theater. Each standard ticket sold allows for 1 attendee, while each premium ticket sold allows for 2 attendees.

Part 1: Create a two-variable linear equation that represents the total number of attendees when all the seats are filled. Clearly define your variables. (5 Points)

Part 2: The theater sells 100 standard tickets and 100 premium tickets. Determine if this ticket sales decision exactly fills the theater. Show your work and state whether the theater is filled with this given scenario. (2 points)

(7 points)

GPT-4o mini · Answer

Let's tackle this problem step-by-step.

Part 1: Create a Two-Variable Linear Equation

Step 1: Define the Variables

Let:

$ S $ = the number of standard tickets sold
$ P $ = the number of premium tickets sold

Step 2: Understanding the Seating Capacity

Each standard ticket allows for 1 attendee.
Each premium ticket allows for 2 attendees.

Step 3: Setting Up the Equation

The total number of attendees, represented by the number of seats filled, can be expressed as: \[ S + 2P = 300 \]

This equation means that the total number of attendees is the sum of attendees from standard tickets and double the attendees from premium tickets, which must equal the total number of seats available (300).

Part 2: Determine if Ticket Sales Fill the Theater

Given Ticket Sales:

Number of standard tickets sold $ S = 100 $
Number of premium tickets sold $ P = 100 $

Step 1: Substitute into the Equation

Substituting the values into our equation: \[ S + 2P = 300 \] \[ 100 + 2(100) = 300 \]

Step 2: Calculate the Total Number of Attendees

Calculating the left side: \[ 100 + 200 = 300 \] \[ 300 = 300 \]

Conclusion

The equation holds true, meaning that the total number of attendees with 100 standard tickets and 100 premium tickets exactly fills the theater.

Summary of Results

Part 1: The linear equation that represents the total number of attendees is: \[ S + 2P = 300 \]
Part 2: With 100 standard tickets and 100 premium tickets, the theater is filled, as the total number of attendees equals the number of available seats (300).

Answer

Dr. Ellison says that the equation y = -3x + 7 has a solution of (2, 13). Is Dr. Ellison right or wrong? (1 point)
Responses

Dr. Ellison is wrong.

More information is needed to answer this question.

Dr. Ellison is right.

GPT-4o mini · Answer

To determine whether Dr. Ellison is correct that the equation $ y = -3x + 7 $ has a solution of $ (2, 13) $, we need to substitute $ x = 2 $ into the equation and check if it results in $ y = 13 $.

Step 1: Substitute $ x $ into the equation.

\[ y = -3(2) + 7 \]

Step 2: Calculate the value of $ y $. \[ y = -6 + 7 = 1 \]

Step 3: Compare the calculated $ y $ with the given point.

The calculated $ y $ value is $ 1 $, not $ 13 $. Therefore, $ (2, 13) $ is not a solution to the equation $ y = -3x + 7 $.

Conclusion

Dr. Ellison is wrong. The correct response is: Dr. Ellison is wrong.

Answer

A satellite tv company charges a one-time installation fee and a monthly service charge. The total cost is modeled by the function y = 99x + 15. Which statement represents the meaning of each part of the function? (1 point)
Responses

y is the total cost, x is the number of months of service, $99 is the installation fee, and $15 is the service charge per month.

x is the total cost, y is the number of months of service, $99 is the installation fee, and $15 is the service charge per month.

y is the total cost, x is the number of months of service, $15 is the installation fee, and $99 is the service charge per month.

x is the total cost, y is the number of months of service, $15 is the installation fee, and $99 is the service charge per month.

GPT-4o mini · Answer

Let's break down the function given, which is:

\[ y = 99x + 15 \]

In this function:

$ y $ represents the total cost.
$ x $ represents the number of months of service.
The coefficient of $ x $ (which is $ 99 $) represents the cost associated with each month of service (the monthly service charge).
The constant term (which is $ 15 $) represents the one-time installation fee.

Now, based on this interpretation, we can clarify the meanings:

$ y $ is the total cost.
$ x $ is the number of months of service.
$99 is the monthly service charge.
$15 is the installation fee.

Given this analysis, the correct response is:

y is the total cost, x is the number of months of service, $99 is the installation fee, and $15 is the service charge per month.

However, it seems there's a mix-up in the description of the $99 and the $15 in the response provided. The corrected interpretations should reflect that $99 is for the monthly charge, and $15 is the installation fee.

Therefore, the accurate interpretation of the function should align with this:

$ y $ is the total cost,
$ x $ is the number of months of service,
$99 is the monthly service charge, and
$15 is the installation fee.

Since none of the provided responses reflect accurate pricing for installation and service charge, none of the statements are entirely correct based on the equation given. The best choice reflecting correct interpretations based on standard conventions would have been written differently.

Answer

What is the rate of change of the table below? x y 1 12 2 19 3 26 4 33 5 40 6 47 (1 point) Responses -12 12 1/7 7

GPT-4o mini · Answer

To find the rate of change from the given table of values, we can look at how much $ y $ changes for a given change in $ x $. The rate of change is typically calculated as the change in $ y $ divided by the change in $ x $.

Let's calculate the change in $ y $ for a change in $ x $:

Using points from the table:

When $ x = 1 $, $ y = 12 $
When $ x = 2 $, $ y = 19 $

Calculating the change from $ x = 1 $ to $ x = 2 $: \[ \Delta y = y(2) - y(1) = 19 - 12 = 7 \] \[ \Delta x = x(2) - x(1) = 2 - 1 = 1 \]

The rate of change for these points is: \[ \text{Rate of Change} = \frac{\Delta y}{\Delta x} = \frac{7}{1} = 7 \]

We can also check the following pairs of values to confirm that the rate of change remains consistent:

From $ x = 2 $ to $ x = 3 $:
- $ \Delta y = 26 - 19 = 7 $
- $ \Delta x = 3 - 2 = 1 $
From $ x = 3 $ to $ x = 4 $:
- $ \Delta y = 33 - 26 = 7 $
- $ \Delta x = 4 - 3 = 1 $
From $ x = 4 $ to $ x = 5 $:
- $ \Delta y = 40 - 33 = 7 $
- $ \Delta x = 5 - 4 = 1 $
From $ x = 5 $ to $ x = 6 $:
- $ \Delta y = 47 - 40 = 7 $
- $ \Delta x = 6 - 5 = 1 $

In each case, the change in $ y $ is consistently $ 7 $ for each increase of $ 1 $ in $ x $, confirming the rate of change is $ 7 $.

Conclusion

The correct response is: 7.

Answer

What is the correct set up to finding the slope of a line that passes through the points (6, −3) and (−2, 7)
?(1 point)
Responses

7−3/−2−6

6−(−3)/−2−7

6−2/−3+7

7−(−3)/−2−6

GPT-4o mini · Answer

To find the slope of a line that passes through two points $(x_1, y_1)$ and $(x_2, y_2)$, we use the formula for the slope $m$:

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

For the points given:

$(x_1, y_1) = (6, -3)$
$(x_2, y_2) = (-2, 7)$

Substituting these points into the slope formula, we have:

\[ m = \frac{7 - (-3)}{-2 - 6} \]

This can be simplified to:

\[ m = \frac{7 + 3}{-2 - 6} = \frac{10}{-8} = -\frac{5}{4} \]

Now, let's compare this setup to the provided options:

Option 1: $ \frac{7 - 3}{-2 - 6} $
- This is incorrect because it does not represent the y-values properly.
Option 2: $ \frac{6 - (-3)}{-2 - 7} $
- This is incorrect because the order of points is mixed up.
Option 3: $ \frac{6 - 2}{-3 + 7} $
- This is incorrect because it does not relate to the points given.
Option 4: $ \frac{7 - (-3)}{-2 - 6} $
- This is correct because it uses the correct values for $y_2$, $y_1$, $x_2$, and $x_1$.

Conclusion

The correct setup to find the slope of the line that passes through the points $ (6, -3) $ and $ (-2, 7) $ is: $ \frac{7 - (-3)}{-2 - 6} $.