Question 1

Question

Question 1

Santiago is working at his mom’s day care this summer. His mom has asked him to check out the library book sale to look for books they might be able to use in the day care. When he gets to the book sale, he sees that board books are $1.50 and soft cover books are $0.50. He picks out a variety of books and spends $20.

Write a two-variable equation to represent this situation. Be sure to define the variables you use in your equation. 
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.  
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. 
What are the constraints of the situation? Describe them. 
Use your graph to help you make a list of all viable solutions to the problem. 
Suppose Santiago bought 34 books. Can you figure out how many of each type of book he bought? Explain. 
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.

Write a two-variable inequality to represent this situation. Be sure to define the variables you use in your inequality. 
Is it possible for the party to include 10 children and five adults? Use your inequality from part A to check. Explain your answer. 
Graph the solution set for the inequality 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. 
What are the constraints of the situation? Describe them. 
Show all the viable solutions to the problem as points on the graph. 
Give a recommendation to Meha as to how many adults and children you think she should invite to the party. Justify your recommendation. 
Question 3

Consider the following system of equations:

−x+y=5y=(x−3)^2 +2

Is  (1,6)
  a solution to the system of equations? 
Explain. Solve the system of equations using substitution.  
Graph the system of equations on a piece of graph paper. On the graph, label the solution(s) to the system of equations. 
Could a system of two equations that includes a linear equation and a quadratic equation ever have an infinite number of solutions? If so, how could you tell from the graph? If not, explain why not. 
Question 4

Consider the following system of equations:

1/3x+y=3
2y=−2/3x + 10
 
Is  (0,3)
  a solution to the system of equations? Explain. 
Solve the system of equations using the elimination method. 
Could a system of two linear equations ever have an infinite number of solutions? If so, how could you tell by looking at their equations? How could you tell from the graph? If not, explain why. 
Question 5

Write your own system of two linear inequalities. 
Describe how you could determine whether a given point is a solution to your system of linear inequalities. Include an example in your description. 
Graph the solution set for the system of linear inequalities on a piece of graph paper. Make sure to clearly indicate where the solution is on the graph. 
Could a system of linear inequalities ever have no solutions? What would this look like on a graph? Explain.

GPT-4o mini · Answer

Let's break down the questions one by one.

Question 1

A) Define the variables:

Let $ x $ be the number of board books.
Let $ y $ be the number of soft cover books.

The cost relationship can be expressed as: \[ 1.5x + 0.5y = 20 \]

B) To check if Santiago bought 10 board books and 5 soft cover books:

Substitute $ x = 10 $ and $ y = 5 $ into the equation: \[ 1.5(10) + 0.5(5) = 15 + 2.5 = 17.5 \] Since $ 17.5 \neq 20 $, it is not possible for him to have bought 10 board books and 5 soft cover books.

C) To graph the equation:

Rearranging the equation: \[ 0.5y = 20 - 1.5x \implies y = 40 - 3x \]
Identify x-intercept (when $ y=0 $): $ x = \frac{40}{3} \approx 13.33 $
Identify y-intercept (when $ x=0 $): $ y = 40 $

D) Constraints of the situation:

$ x \geq 0 $ (cannot buy negative books)
$ y \geq 0 $ (cannot buy negative books)
Since he cannot spend more than $20, the equation reflects that as well.

E) Viable solutions:

The graph will show a line with the positive quadrant below it, where the solutions must satisfy the non-negativity constraints.

F) If Santiago bought 34 books: \[ x + y = 34 \] We have the system:

$ 1.5x + 0.5y = 20 $
$ x + y = 34 $

Substituting $ y = 34 - x $ into the first equation gives: \[ 1.5x + 0.5(34 - x) = 20 \] Solving for $ x $: \[ 1.5x + 17 - 0.5x = 20 \implies x = 6 \] Using $ x + y = 34 $: \[ 6 + y = 34 \implies y = 28 \] Thus, 6 board books and 28 soft cover books.

Question 2

A) Define the variables:

Let $ c $ be the number of children.
Let $ a $ be the number of adults.

The total cost inequality: \[ 10.50c + 5.50a + 150 \leq 300 \]

B) To check 10 children and 5 adults: \[ 10.50(10) + 5.50(5) + 150 = 105 + 27.5 + 150 = 282.5 \] Since $ 282.5 \leq 300 $, it is possible.

C) Graphing the inequality:

Rearrange to get: \[ 10.50c + 5.50a \leq 150 \]

D) Constraints of the situation:

$ c \geq 1 $ (at least one child)
$ a \geq 1 $ (at least one adult)
Spending limit of $300.

E) Viable solutions will be shaded below the line on the graph, constrained to positive $ c $ and $ a $.

F) Recommendation for invitations will depend on finding a reasonable balance while keeping below the budget and satisfying the minimum count constraints.

Question 3

A) Determine if $ (1,6) $ is a solution: \[ -x + y = 5 \implies -1 + 6 = 5 \quad \text{(True)} \] Substituting into the second equation: \[ y = (x - 3)^2 + 2 \implies 6 = (1 - 3)^2 + 2 \implies 6 = 4 + 2 \quad \text{(True)} \] Thus, $ (1,6) $ is a solution.

B) To solve using substitution: Setting $ y = (x - 3)^2 + 2 $:

Substitute in $-x + y = 5$, \[ -x + (x - 3)^2 + 2 = 5 \] Rearranged gives you a polynomial to solve.

C) Graphing the equations can be done, observing where they intersect gives the solution.

D) For a linear and quadratic equation, they could intersect once, twice, or not at all, indicating finite solutions, so infinite solutions are not possible.

Question 4

A) Checking if $ (0,3) $ is a solution: For the first equation: \[ \frac{1}{3}(0) + 3 = 3 \quad \text{(True)} \] For the second equation: \[ 2(3) = -\frac{2}{3}(0) + 10 \quad \text{(True)} \] So $ (0,3) $ is a solution.

B) Using elimination to solve:

Multiply the first equation to eliminate fractions.
Combine equations to solve for $x$ or $y$.

C) Systems of linear equations can be parallel (no solution), intersect in a point (one solution), or be the same line (infinitely many solutions), which can be analyzed via coefficients.

Question 5

A) Example system:

$ y \geq 2x + 1 $
$ y < -x + 4 $

B) A point (for example, (0,3)) is tested by substituting into both inequalities. If the point satisfies both, it is a solution.

C) The graph would show shaded regions indicating solutions, where the two areas overlap.

D) A system can have no solution if lines are parallel, showing distinct, non-intersecting paths on a graph.

These are the structured answers for the questions you provided based on the given scenarios in mathematics particularly associated with linear equations and inequalities. If you still need further breakdown, just ask!