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A. Write a two-variable equation to represent this situation. Be sure to define the variables

you use in your equation.

11 answers

To write a two-variable equation to represent a situation, I will need some context or details about the specific scenario you have in mind. However, I can provide you with a general format and example.

Example Situation:

Let's say we have a situation involving money spent on two types of fruit: apples and bananas.

Definition of Variables:

  • Let \( x \) represent the number of apples purchased.
  • Let \( y \) represent the number of bananas purchased.

Assumption:

Assume apples cost $2 each and bananas cost $1 each. If the total amount spent on fruit is $10, we can write an equation to represent this situation.

Equation:

The equation representing the situation can be stated as follows:

\[ 2x + 1y = 10 \]

This means that the cost of apples (2 times the number of apples) plus the cost of bananas (1 times the number of bananas) equals the total amount spent ($10).

Please provide specific details if you need a tailored equation for a different situation!

B. Is it possible that Santiago bought 10 board books and five soft cover books? Use your

equation from part A to check. Explain your answer.

To determine if it's possible that Santiago bought 10 board books and five soft cover books, we first need to define our variables and their associated costs.

Definition of Variables:

  • Let \( b \) represent the number of board books purchased.
  • Let \( s \) represent the number of soft cover books purchased.

Assumption:

Let's assume:

  • Each board book costs $5.
  • Each soft cover book costs $3.

Total Cost Equation:

We can write the total cost equation based on the numbers and costs of the books as follows:

\[ 5b + 3s = T \]

Where \( T \) is the total amount spent.

Given Information:

In this case, Santiago bought:

  • \( b = 10 \) (board books)
  • \( s = 5 \) (soft cover books)

Substituting Values:

Now we'll substitute these values into our equation to find \( T \):

\[ 5(10) + 3(5) = T \]

Calculating that:

\[ 50 + 15 = T \]

\[ T = 65 \]

Conclusion:

Thus, if Santiago bought 10 board books and 5 soft cover books, he would have spent a total of $65.

Is it possible? Yes, it is definitely possible for Santiago to have bought 10 board books and five soft cover books at these prices, as indicated by the calculation.

If you had different prices or totals in mind, feel free to provide that information, and I can assist further!

C. Graph the solution set for the equation you wrote in part A on a piece of graph paper. Note:

At this point, do not yet consider any constraints given by the situation.

While I can't create an image of a graph for you directly, I can guide you through the process of graphing the equation from part A, which was:

\[ 2x + 1y = 10 \]

Steps to Graph the Equation

  1. Rearrange the Equation: Rewrite the equation in slope-intercept form (y = mx + b):

    \[ y = -2x + 10 \]

    Here, the slope (m) is -2, and the y-intercept (b) is 10.

  2. Identify Points: To graph the equation, find two points that satisfy the equation:

    • Y-intercept: Set \( x = 0 \) \[ y = -2(0) + 10 = 10 \quad \text{(point: (0, 10))} \]

    • X-intercept: Set \( y = 0 \) \[ 0 = -2x + 10 \implies 2x = 10 \implies x = 5 \quad \text{(point: (5, 0))} \]

  3. Plot Points: On graph paper, plot the points (0, 10) and (5, 0).

  4. Draw the Line: Use a ruler to draw a straight line through the points you plotted. This line represents the solution set for the equation \( 2x + y = 10 \).

  5. Label the Axes: Don’t forget to label your axes: the horizontal axis as \( x \) and the vertical axis as \( y \).

Additional Information:

  • The line you’ve drawn extends infinitely in both directions. Since we haven’t added any constraints yet, any point on this line is a solution to the equation.

  • You can also mark additional points on the line to check your graph's accuracy. For example, if you pick \( x = 1 \):

    \[ y = -2(1) + 10 = 8 \quad \text{(point: (1, 8))} \]

    Plot the point (1, 8) to see that it lies on the line you have drawn.

This process will create a visual representation of the solution set for your two-variable equation. If you have any further question or need help with constraints, feel free to ask!

E. Use your graph to help you make a list of all viable solutions to the problem.

To identify viable solutions to the problem using the graph of the equation \( 2x + y = 10 \), we need to consider both the graph and the context of the situation. However, since we haven't specified any constraints yet, let's summarize how you can determine viable solutions based on the graph and context.

Context Consideration

Assuming we are still focusing on the previously established scenario of purchasing board books and softcover books, let's reaffirm the variables:

  • \( x \): the number of board books
  • \( y \): the number of softcover books

Solutions from the Graph

From the graph of the equation \( y = -2x + 10 \), we know that:

  • Any point \((x, y)\) on the line is a solution to the equation.
  • Since we are talking about purchasing books, both \( x \) (number of board books) and \( y \) (number of softcover books) should be non-negative integers, as you can't buy a negative quantity of books.

Finding Viable Solutions

To find viable solutions, we can examine the integer points along the line \( 2x + y = 10 \) while ensuring both \( x \) and \( y \) are non-negative.

  1. Solve for Integer Solutions: We can incrementally try integer values for \( x \) and solve for \( y \):

    • If \( x = 0 \): \[ y = 10 - 2(0) = 10 \quad \text{(solution: (0, 10))} \]

    • If \( x = 1 \): \[ y = 10 - 2(1) = 8 \quad \text{(solution: (1, 8))} \]

    • If \( x = 2 \): \[ y = 10 - 2(2) = 6 \quad \text{(solution: (2, 6))} \]

    • If \( x = 3 \): \[ y = 10 - 2(3) = 4 \quad \text{(solution: (3, 4))} \]

    • If \( x = 4 \): \[ y = 10 - 2(4) = 2 \quad \text{(solution: (4, 2))} \]

    • If \( x = 5 \): \[ y = 10 - 2(5) = 0 \quad \text{(solution: (5, 0))} \]

  2. List of Viable Solutions: Based on this analysis, the list of viable solutions that meet the conditions (non-negative integers) is:

    • (0, 10): 0 board books and 10 softcover books
    • (1, 8): 1 board book and 8 softcover books
    • (2, 6): 2 board books and 6 softcover books
    • (3, 4): 3 board books and 4 softcover books
    • (4, 2): 4 board books and 2 softcover books
    • (5, 0): 5 board books and 0 softcover books

Conclusion

These points represent all the combinations of board books and softcover books Santiago could have purchased while spending a total of $10 under the provided pricing assumptions. If you have any additional constraints or a different scenario in mind, please provide those details for further assistance!

F. Suppose Santiago bought 34 books. Can you figure out how many of each type of book he

bought? Explain.

To determine how many of each type of book Santiago bought if he purchased a total of 34 books, we can set up a system of equations using the variables we defined earlier:

Definitions of Variables:

  • Let \( b \) represent the number of board books.
  • Let \( s \) represent the number of softcover books.

Given Information:

  1. Total Cost Equation: From earlier, we know: \[ 2b + s = 10 \] This was from the cost information provided initially (assuming board books were $5 each and softcover books were $3 each).

  2. Total Number of Books: We also know: \[ b + s = 34 \] This tells us that the total number of books Santiago bought is 34.

Set Up the System of Equations:

Now, we have the following system:

  1. \( 2b + s = 10 \) (Cost Equation)
  2. \( b + s = 34 \) (Total Books Equation)

Solving the System of Equations:

We can solve this system using substitution or elimination. Let's use the elimination method.

Step 1: Solve the second equation for \( s \): \[ s = 34 - b \]

Step 2: Substitute \( s \) into the first equation: \[ 2b + (34 - b) = 10 \]

Step 3: Simplify and solve for \( b \): \[ 2b + 34 - b = 10 \] \[ b + 34 = 10 \] \[ b = 10 - 34 \] \[ b = -24 \]

Since \( b \) cannot be negative, this indicates an inconsistency in our equations based on the information provided.

Reevaluating the Data

  1. Given the cost constraint with a maximum total of $10, it is impossible to buy 34 books under these conditions since even the minimum (which would necessitate a higher total than $10) exceeds that amount.

Conclusion

Based on the equations and the context, it's evident that without additional information or corrections to the pricing or constraints, it's not possible for Santiago to have bought a total of 34 books while also adhering to the cost limitation of $10.

If there are further details or corrections to be provided (e.g., different prices or no cost limit), that information would be necessary to reassess the problem.

Question 2

Meha is helping her aunt to plan a birthday party for her cousin at the science center. Her aunt

tells her that the party will cost $150, plus $10.50 per child and $5.50 per adult. There needs to be at least one adult and at least one child at the party. She has budgeted $300 for the party and that is the maximum amount she wants to spend.

A. Write a two-variable inequality to represent this situation. Be sure to define the variables

you use in your inequality.

To represent this situation with a two-variable inequality, we need to define our variables and express the constraints based on the details provided.

Definitions of Variables:

  • Let \( c \) represent the number of children attending the party.
  • Let \( a \) represent the number of adults attending the party.

Cost Structure:

The total cost of the party is composed of:

  • A fixed cost of $150.
  • A variable cost of $10.50 per child, giving a total cost for children of \( 10.50c \).
  • A variable cost of $5.50 per adult, giving a total cost for adults of \( 5.50a \).

Total Cost Inequality:

The total cost of the party can therefore be expressed as: \[ \text{Total Cost} = 150 + 10.50c + 5.50a \]

The total cost must be less than or equal to the budgeted maximum amount of $300. This leads to the inequality: \[ 150 + 10.50c + 5.50a \leq 300 \]

Simplifying the Inequality:

We can simplify this inequality: \[ 10.50c + 5.50a \leq 300 - 150 \] \[ 10.50c + 5.50a \leq 150 \]

Additional Constraints:

According to the problem, there needs to be at least one adult and at least one child at the party. Hence, we have the additional constraints: \[ c \geq 1 \] \[ a \geq 1 \]

Summary:

In summary, the two-variable inequality representing the situation is:

  1. Main Inequality: \[ 10.50c + 5.50a \leq 150 \]
  2. Constraints: \[ c \geq 1 \] \[ a \geq 1 \]

This captures the cost constraints and the required presence of at least one adult and one child at the party. If you have any further questions or need assistance with later parts of the problem, feel free to ask!