To solve this system, you could do the following steps:

Question

3x+4y=48

6x+2y=60

(6 points)
1. Choose the best method to solve by looking at the way the equations are set up for you: (Response area)

2. Using the best method, how could you get the x variables to cancel out? (Response area)

3. After you complete step 2, what would your top equation be? (Response area)

4. Line them up and add the equations together to get this equation. (Response area)

5. Solve for y. (Response area)

6. Solve for x. (Response area)
responses
Choose a method
substitution
graphing
elimination

Operation to get the x variables to cancel
multiply the bottom equation x 2
multiply the top equation x -2
divide the top equation by 4
divide the bottom equation by 2

Simplified top equation
x + 2y = 46
-6x - 8y = -96
3x + y = 30
-6x + 4y = 96

After adding the equations
-6y = -36
6y = -96
-8x = -96
8y = 96
Solve for x and y
y= -12
x= -3
y = 6
y = 18
x= 8
x= 6

Answer 1

Let's solve the given system of equations step by step as per your instructions.

Given Equations:

1. \( 3x + 4y = 48 \) (Equation 1)
2. \( 6x + 2y = 60 \) (Equation 2)

1. Method of Choice:

Choose a method: elimination
This method is suitable since the equations are set up so that we can easily manipulate them to cancel out one variable.

2. Getting the x variables to cancel out:

Operation to get the x variables to cancel:
Multiply the top equation by -2 so that the \( x \) terms in both equations will cancel out when added: \[ -2(3x + 4y = 48) \Rightarrow -6x - 8y = -96 \] Now, we can use this to cancel with the second equation.

3. Simplified top equation:

The simplified top equation from the operation is:
\(-6x - 8y = -96\)

4. Line them up and add the equations together:

Now we need to line up the modified top equation with the bottom equation: \[ \begin{align*} -6x - 8y & = -96 \quad \text{(Modified Equation 1)} \ 6x + 2y & = 60 \quad \text{(Equation 2)} \ \end{align*} \]

Adding these two equations together: \[ (-6x + 6x) + (-8y + 2y) = -96 + 60 \] This simplifies to: \[ 0x - 6y = -36 \]

5. Solve for y:

To solve for \( y \): \[ -6y = -36 \ y = \frac{-36}{-6} = 6 \]

6. Solve for x:

Now we can substitute \( y = 6 \) back into one of the original equations, let's use Equation 1: \[ 3x + 4(6) = 48 \ 3x + 24 = 48 \ 3x = 48 - 24 \ 3x = 24 \ x = \frac{24}{3} = 8 \]

Final Solution:

\( x = 8 \) and \( y = 6 \).

Thus, the solution to the system of equations is \( (x, y) = (8, 6) \).

Answer 2

The Johnsons and the Wilsons went to the Burger Barn for dinner.

The Johnsons bought 2 burger meals and 5 hot dog meals for $46.

The Wilsons bought 5 burger meals and 2 hot dog meals for $52.

When you solve, you get a solution of (8, 6).

(3 points)
What does the 8 mean in context?
What does the 6 mean in context?
Who got the better deal?

responses
how many burger meals
how many hot dog meals
the cost of a burger meal
the cost of a hot dog meal
the total cost of a burger + hot dog meals
neither
both
the Johnson's
the Wilson's

Answer 3

To analyze the situation, let's define the variables based on the context of the problem:

Let:

• \( b \) = cost of a burger meal
• \( h \) = cost of a hot dog meal

Given that the solution to the system resulted in \( (8, 6) \):

• The first number, 8, corresponds to the cost of a burger meal.
• The second number, 6, corresponds to the cost of a hot dog meal.

Now we can answer your questions based on this interpretation:

Responses:

1. What does the 8 mean in context?

   • the cost of a burger meal

2. What does the 6 mean in context?

   • the cost of a hot dog meal

3. Who got the better deal?

   • We need to look at the total cost for each family and see who spent less.

Calculating the total cost for the Johnsons and the Wilsons:

• The Johnsons bought 2 burger meals and 5 hot dog meals: \[ \text{Total cost for Johnsons} = 2(8) + 5(6) = 16 + 30 = 46 \text{ dollars} \]

• The Wilsons bought 5 burger meals and 2 hot dog meals: \[ \text{Total cost for Wilsons} = 5(8) + 2(6) = 40 + 12 = 52 \text{ dollars} \]

Since the Johnsons spent $46 and the Wilsons spent $52, we conclude that:

• the Johnson's got the better deal.

Answer 4

Does the line through (3, 7) and (-2, -5) intersect with the line through (4, 8) and (10, -2)?(1 point)
Responses

No…these lines are perpendicular
No…these lines are perpendicular

No…these lines are parallel
No…these lines are parallel

Yes…these lines are not parallel
Yes…these lines are not parallel

Yes…these lines are parallel
Yes…these lines are parallel

You cannot tell without a graph
You cannot tell without a graph

Yes…these lines are perpendicular

Answer 5

To determine if the lines intersect, we'll first find the slopes of both lines. If the slopes are equal, the lines are parallel (and won't intersect). If the slopes are negative reciprocals of each other, the lines are perpendicular (and intersect). If neither condition is true, the lines will intersect at some point.

Line through (3, 7) and (-2, -5):

The formula for the slope \( m \) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:

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

Substituting in the points (3, 7) and (-2, -5):

( x_1 = 3, y_1 = 7 \ x_2 = -2, y_2 = -5 )

\[ m_1 = \frac{-5 - 7}{-2 - 3} = \frac{-12}{-5} = \frac{12}{5} \]

Line through (4, 8) and (10, -2):

Using the same formula for the points (4, 8) and (10, -2):

( x_1 = 4, y_1 = 8 \ x_2 = 10, y_2 = -2 )

\[ m_2 = \frac{-2 - 8}{10 - 4} = \frac{-10}{6} = -\frac{5}{3} \]

Summary of Slopes:

• Slope of the first line: \( m_1 = \frac{12}{5} \)
• Slope of the second line: \( m_2 = -\frac{5}{3} \)

Conclusion:

The slopes are different (\( \frac{12}{5} \) is not equal to \(-\frac{5}{3}\)), which means the lines are not parallel. Since their slopes are also not negative reciprocals (i.e., they are not perpendicular), we can conclude that the lines do intersect at some point.

Final Response:

Yes…these lines are not parallel.

Answer 6

An amusement park that you are excited to try is $28 per visit.

You have the option to purchase a monthly membership for $90 and then pay $15 for each visit.

(4 points)
You have plans to attend 3 times this summer. Which option is best?
You think you will visit probably about 8 times this summer. Which option is best?
You have budgeted $150 for visiting the park this summer. Which option is best?
How many visits would be approximately the break even point where both options would cost about the same?

responses
pay per visit
buy a membership
2
5
7
11

Answer 7

Let's calculate the total costs for both options in different scenarios, where we compare the pay-per-visit option with the membership option.

1. Cost Calculation for Each Option

• Pay-per-visit cost: Cost = $28 per visit \(\times\) number of visits

• Membership cost: Cost = $90 + $15 per visit \(\times\) number of visits

A. Plans to Attend 3 Times This Summer:

Pay-per-visit cost: \[ \text{Cost} = 28 \times 3 = 84 \text{ dollars} \]

Membership cost: \[ \text{Cost} = 90 + 15 \times 3 = 90 + 45 = 135 \text{ dollars} \]

Best option:

• Pay per visit ($84) is better than buying a membership ($135).

B. Think You Will Visit About 8 Times This Summer:

Pay-per-visit cost: \[ \text{Cost} = 28 \times 8 = 224 \text{ dollars} \]

Membership cost: \[ \text{Cost} = 90 + 15 \times 8 = 90 + 120 = 210 \text{ dollars} \]

Best option:

• Buy a membership ($210) is better than pay per visit ($224).

C. Have Budgeted $150 for Visiting the Park This Summer:

Pay-per-visit cost: \[ \text{Cost} = 28 \times n \text{ (where \( n \) is the number of visits)} \]

Membership cost: \[ \text{Cost} = 90 + 15 \times n \]

Now we need to find the maximum number of visits for each option to stay within a $150 budget.

• For pay-per-visit: \[ 28n \leq 150 \Rightarrow n \leq \frac{150}{28} \approx 5.36 \Rightarrow n = 5 \text{ visits} \]

  The cost for 5 visits: \[ 28 \times 5 = 140 \text{ dollars} \]

• For membership: \[ 90 + 15n \leq 150 \Rightarrow 15n \leq 60 \Rightarrow n \leq \frac{60}{15} = 4 \]

  The cost for 4 visits: \[ 90 + 15 \times 4 = 90 + 60 = 150 \text{ dollars} \]

Best option:

• Pay per visit allows for 5 visits at $140, whereas the membership only allows for 4 visits at $150.

D. Break-even Point Calculation:

To find the break-even point where both options cost the same:

Set the costs equal: \[ 28n = 90 + 15n \]

Rearranging gives: \[ 28n - 15n = 90 \ 13n = 90 \ n = \frac{90}{13} \approx 6.92 \]

So, the break-even point is approximately 7 visits.

Summary of Best Options:

• For 3 visits: Pay per visit is best.
• For 8 visits: Buy a membership is best.
• With a budget of $150: Pay per visit is the best option.

Final Response:

• You have plans to attend 3 times: pay per visit
• You think you will visit probably about 8 times: buy a membership
• You have budgeted $150 for visiting the park this summer: pay per visit
• How many visits would be approximately the break-even point: 7