Where indicated, draw figures on a blank sheet of paper or on a sheet of graph paper. Be sure to label each sheet so that your teacher knows which answer goes with which question. You can scan these figures and submit them as individual documents with your portfolio worksheet, or you can take pictures of them and insert the pictures on the portfolio worksheet.

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:

13x+y=32y=−23x+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

Sure! Let’s go through each question step by step.

Question 1

A. Write a Two-Variable Equation:

Let:

$ x $ = the number of board books
$ y $ = the number of soft cover books

The equation to represent the spending is: \[ 1.50x + 0.50y = 20 \]

B. Check for 10 Board Books and 5 Soft Cover Books:

Substituting $ x = 10 $ and $ y = 5 $ into the equation: \[ 1.50(10) + 0.50(5) = 15 + 2.5 = 17.5 \]

This does not equal $20, so it is not possible for Santiago to have bought 10 board books and 5 soft cover books.

C. Graph the Solution Set:

To graph the equation, rearrange it to slope-intercept form $ y = mx + b $: \[ 0.50y = 20 - 1.50x \]
\[ y = -3x + 40 \]

On graph paper, plot this line, marking the x-intercept and y-intercept.

D. Describe the Constraints:

The constraints of this situation include:

$ x \geq 0 $ (number of board books cannot be negative)
$ y \geq 0 $ (number of soft cover books cannot be negative)

E. List of Viable Solutions:

From the graph, for integer solutions, check the intersection points within the first quadrant defined by constraints.

F. Determine the Number of Each Type of Book if Santiago Bought 34 Books:

Set up the new equation: \[ x + y = 34 \] Now you have a system of equations:

$ 1.50x + 0.50y = 20 $
$ x + y = 34 $

You can solve this system to find the values of $ x $ and $ y $.

Question 2

A. Write a Two-Variable Inequality:

Let:

$ c $ = number of children
$ a $ = number of adults

The inequality to represent the cost is: \[ 150 + 10.50c + 5.50a \leq 300 \]

B. Check for 10 Children and 5 Adults:

Substituting $ c = 10 $ and $ a = 5 $: \[ 150 + 10.50(10) + 5.50(5) = 150 + 105 + 27.5 = 282.5 \leq 300 \]

This is possible!

C. Graph the Solution Set:

Rearranging the inequality to slope-intercept form: \[ 10.50c + 5.50a \leq 150 \]

Graph using intercepts determined from the equation.

D. Describe the Constraints:

$ c \geq 1 $ (at least one child)
$ a \geq 1 $ (at least one adult)

E. Show All Viable Solutions on the Graph:

Mark solutions within the constraints on your graph.

F. Recommendation to Meha:

Based on the budget and constraints, a balanced number of adults and children (e.g., 10 children and 5 adults) could be suggested since it fits within budget.

Question 3

A. Is (1,6) a Solution?

Substituting into both equations:

For the first equation: $-1 + 6 = 5$ ✔️
For the second equation: $6 = (1-3)^2 + 2 \rightarrow 6 = 4 + 2$ ❌

So, (1,6) is not a solution.

B. Solve the System using Substitution:

From the first equation, express $ y $: \[ y = 5 + x \]

Substituting into the second, solve for $ x $.

C. Graph the System of Equations:

Once solved, graph both equations and label intersection points.

D. Infinite Solutions with a Linear and Quadratic:

Yes, if the quadratic just touches the line (tangential intersection).

Question 4

A. Is (0,3) a Solution?

Substituting into both equations:

For the first: $0 + 3 = 32$ ❌
For the second: $3 = -2 \times 0 + 10$ ✔️

(0,3) is not a solution.

B. Solve using Elimination:

Align both equations and solve for $ x $ and $ y $ through elimination.

C. Infinite Solutions with Two Linear Equations:

Yes, two linear equations can overlap entirely, indicating infinite solutions.

Question 5

A. Write a System of Two Linear Inequalities:

$ y \leq 2x + 3 $
$ y > -x + 4 $

B. Determine if Point is a Solution:

Test a point (e.g., (1, 4)):

For $ y \leq 2x + 3 $ → $ 4 \leq 2(1) + 3 → 4 \leq 5 $ ✔️
For $ y > -x + 4 $ → $ 4 > -1 + 4 → 4 \not> 3 $ ❌

C. Graph the Inequalities:

Graph the lines and shade regions as per the inequalities.

D. System with No Solutions:

Yes, parallel lines indicate no solution. No intersection on a graph would demonstrate this.

These steps should guide you in answering each part of the questions with the necessary definitions, explanations, and graphs! Depending on your capabilities, you may need the graphs plotted on graph paper as required.

Answer

rethink qeustion A part E

GPT-4o mini · Answer

Certainly! Let's reassess Question 1, Part E—where we're looking to identify a list of viable solutions based on the graph of the equation we set up earlier.

Question 1, Part E: List of All Viable Solutions

For our two-variable equation $1.50x + 0.50y = 20$, we determined:

$ x = $ number of board books,
$ y = $ number of soft cover books.

Identifying Viable Solutions:

Constraints Recap:
- Both $ x $ and $ y $ must be non-negative integers (since you can't purchase a negative number of books or a fraction of a book).
Finding Integer Solutions: From the equation $1.50x + 0.50y = 20$, let's manipulate it to isolate one variable in terms of the other:

\[ 0.50y = 20 - 1.50x \]

Dividing everything by 0.50 (to clear the decimal): \[ y = 40 - 3x \]

This means for each value of $ x $, we can find a corresponding $ y $ by substituting into this equation.

Solutions:

Since both $ x $ and $ y $ must be non-negative:
- Set $ y \geq 0 $:
\[ 40 - 3x \geq 0 \implies x \leq \frac{40}{3} \approx 13.33 \]

Since $ x $ must be an integer, the maximum value for $ x $ is 13.
Now, substituting possible integer values for $ x $ from 0 to 13 and calculating $ y $:

| $ x $ | $ y = 40 - 3x $ | Viable Solution | |---------|------------------|-------------------| | 0 | 40 | (0, 40) | | 1 | 37 | (1, 37) | | 2 | 34 | (2, 34) | | 3 | 31 | (3, 31) | | 4 | 28 | (4, 28) | | 5 | 25 | (5, 25) | | 6 | 22 | (6, 22) | | 7 | 19 | (7, 19) | | 8 | 16 | (8, 16) | | 9 | 13 | (9, 13) | | 10 | 10 | (10, 10) | | 11 | 7 | (11, 7) | | 12 | 4 | (12, 4) | | 13 | 1 | (13, 1) |

Summary of Viable Solutions:

The viable integer solutions for $ (x, y) $ within this context, meaning the combinations of board books and soft cover books Santiago could buy while spending exactly $20, are:

(0, 40)
(1, 37)
(2, 34)
(3, 31)
(4, 28)
(5, 25)
(6, 22)
(7, 19)
(8, 16)
(9, 13)
(10, 10)
(11, 7)
(12, 4)
(13, 1)

This re-evaluation gives a clearer and structured list based on the equation we derived!

Answer

rethink question 2 parts c and e

GPT-4o mini · Answer

Sure! Let’s re-evaluate Question 2, specifically Parts C and E, focusing on the inequalities representing Meha's birthday party budget situation.

Question 2

Part C: Graph the Solution Set for the Inequality

A. Recap of the Inequality: We previously established the inequality based on the party costs: \[ 150 + 10.50c + 5.50a \leq 300 \] This can be simplified to: \[ 10.50c + 5.50a \leq 150 \]

B. Rewrite the Inequality: To graph, we rearrange to find $ a $ in terms of $ c $: \[ 5.50a \leq 150 - 10.50c \] \[ a \leq \frac{150 - 10.50c}{5.50} \] \[ a \leq \frac{150}{5.50} - \frac{10.50}{5.50}c \] Calculating those values: \[ a \leq 27.27 - 1.91c \]

Identify Intercepts:
- When $ c = 0 $: \[ a \leq 27.27 \] (y-intercept is about 27.27)
- When $ a = 0 $: \[ 0 \leq 27.27 - 1.91c \] \[ 1.91c \leq 27.27 \implies c \leq 14.27 \] (x-intercept is about 14.27)
Plotting the Inequality:
- Graph the line with endpoint at $ (0, 27.27) $ and $ (14.27, 0) $.
- Shade below the line to represent $ a \leq 27.27 - 1.91c $.
Mark Constraints:
- Remember that we also have constraints of at least one adult $ (a \geq 1) $ and one child $ (c \geq 1) $.

Part E: Show All Viable Solutions on the Graph

Constraints Recap:
- $ c \geq 1 $ (at least 1 child)
- $ a \geq 1 $ (at least 1 adult)
- Budget constraint graphically represented by the shaded area below the line we created.
Identifying Viable Solutions:
- Look for points in the shaded area of the graph that also satisfy the constraint $ c \geq 1 $ and $ a \geq 1 $.

3. Sample Points to Consider:

Viable solutions can include:
- $ (1, 20) $: 1 child, 20 adults
- $ (5, 15) $: 5 children, 15 adults
- $ (10, 5) $: 10 children, 5 adults
- $ (12, 1) $: 12 children, 1 adult

Recommendation to Meha: Based on the viable solutions shown on the graph:
- Recommended Numbers: Invite 5 children and 15 adults.
- Justification: This setup stays well within the budget of $300, while also allowing for a reasonable number of attendees, ensuring that at least one child and adult are present. You can adjust the numbers while still staying within the solution region on the graph.

Summary

In conclusion, reanalyzing Part C, we properly clarified the graphical steps needed to represent the inequalities, while in Part E, we highlighted viable solutions based not only on the budget but also on the constraints that ensure both minimum attendance of children and adults at the party.

Answer

so what are the recommended graph chart what is the recommended numbers on it what should i use?

GPT-4o mini · Answer

To clearly represent your findings in Question 2, here’s a suggested approach for creating the recommended graph, detailing how to indicate viable solutions, and culminating in the selection of recommended numbers for Meha’s birthday party.

Conclusion

When you create and label your graph, include the feasible region, the plotted viable points, and clearly indicate your recommendation of 5 children and 15 adults. These steps facilitate a comprehensive visual representation of how the decision was arrived at while ensuring clarity in your recommendations to Meha concerning the birthday party planning.

Graphing Tools

You can create this graph using graph paper, drawing tools, or graphing software, such as GeoGebra, Desmos, or on paper.
If submitting digitally, consider taking a clear picture of your graph and attaching it to your portfolio worksheet.

Answer

rethink all parts of question three expand them.

GPT-4o mini · Answer

Certainly! Let’s thoroughly expand each part of Question 3 regarding the given system of equations, ensuring clarity and detail in each explanation and step.

Question 3

Given the system of equations:

$-x + y = 5$
$y = (x - 3)^2 + 2$

A. Is $ (1, 6) $ a Solution to the System of Equations?

To determine if $ (1, 6) $ is a solution, substitute $ x = 1 $ and $ y = 6 $ into both equations:

For the first equation: \[ -1 + 6 = 5 \] \[ 5 = 5 \quad \text{(True)} \]
For the second equation: \[ 6 = (1 - 3)^2 + 2 \] \[ 6 = (-2)^2 + 2 \] \[ 6 = 4 + 2 \] \[ 6 = 6 \quad \text{(True)} \]

Since both equations are satisfied, $ (1, 6) $ is a solution to the system of equations.

B. Solve the System of Equations Using Substitution

Since we already have $ y $ expressed in the second equation, we can substitute this expression back into the first equation.

Start with the first equation: \[ -x + y = 5 \]
Substituting for $ y $: \[ -x + (x - 3)^2 + 2 = 5 \] Rearranging gives: \[ (x - 3)^2 - x + 2 = 5 \] \[ (x - 3)^2 - x - 3 = 0 \]
Expanding $(x - 3)^2$: \[ x^2 - 6x + 9 - x - 3 = 0 \] Simplifying: \[ x^2 - 7x + 6 = 0 \]
Factoring the quadratic equation: \[ (x - 1)(x - 6) = 0 \] The solutions are: \[ x = 1 \quad \text{or} \quad x = 6 \]
Finding the corresponding $ y $ values:
- For $ x = 1 $: \[ y = (1 - 3)^2 + 2 = 4 + 2 = 6 \quad \Rightarrow (1, 6) \]
- For $ x = 6 $: \[ y = (6 - 3)^2 + 2 = 9 + 2 = 11 \quad \Rightarrow (6, 11) \]

Solutions: The system has two solutions: $ (1, 6) $ and $ (6, 11) $.

C. Graph the System of Equations

Graphing the linear equation $-x + y = 5$:
- Rearranging it to slope-intercept form:
  \[ y = x + 5 \]
- Identify the y-intercept (0, 5) and the x-intercept (5, 0). Plot these points and draw the line.
Graphing the quadratic equation $y = (x - 3)^2 + 2$:
- Identify the vertex: The vertex form tells us the vertex is at $ (3, 2) $.
- Find additional points by choosing x-values.
  - $ x = 2: y = (2-3)^2 + 2 = 1 + 2 = 3 \Rightarrow (2, 3) $
  - $ x = 4: y = (4-3)^2 + 2 = 1 + 2 = 3 \Rightarrow (4, 3) $
  - $ x = 1: y = 6 $ and $ x = 6: y = 11 $
- Plot the vertex and these points to draw the upward-opening parabola.
Marking the Solutions:
- Identify and label the points where the two graphs intersect: $ (1, 6) $ and $ (6, 11) $.
- Shade the area where the solutions reside.

D. Does a System of Two Equations that Includes a Linear Equation and a Quadratic Equation Ever Have an Infinite Number of Solutions?

Understanding the nature of intersections:
- A linear equation (a straight line) and a quadratic equation (a parabola) can intersect in different ways:
  - At no points (they do not touch).
  - At one point (they touch at one point or are tangent).
  - At two distinct points (the line intersects with the parabola at two distinct locations).
Infinite Solutions Condition:
- A system can have infinite solutions only if the linear equation is the same as the quadratic equation, which means that the line can be rewritten to match the quadratic.
Graphical Representation:
- If the straight line overlaps with the parabola completely (which would mean they share all infinite points), they must be equal in their equations—this overlap represents infinite solutions.

Summary

The detailed examination of each part of Question 3 enables a comprehensive understanding of the proposed system of equations. You explored verifying a solution, solving via substitution, graphing both equations along with their intersections, and analyzing the potential for infinite solutions. Each part delves into both analytical methods and graphing techniques, ensuring clear communication of the mathematical concepts involved.

Answer

now do the same with all the parts of question 4

GPT-4o mini · Answer

Absolutely! Let's go through Question 4 in detail, breaking down each part for clarity and depth regarding the system of equations.

Question 4

Given the system of equations:

$ 13x + y = 32 $
$ y = -23x + 10 $

A. Is $ (0, 3) $ a Solution to the System of Equations?

To determine if $ (0, 3) $ is a solution to the system, we will substitute $ x = 0 $ and $ y = 3 $ into both equations:

For the first equation: \[ 13(0) + 3 = 32 \] \[ 3 = 32 \quad \text{(False)} \]
For the second equation: \[ 3 = -23(0) + 10 \] \[ 3 = 10 \quad \text{(False)} \]

Since both calculations are incorrect, $ (0, 3) $ is not a solution to the system of equations.

B. Solve the System of Equations Using the Elimination Method

To solve the system using elimination, we can manipulate the equations so we can eliminate one variable.

Starting equations: \[ 13x + y = 32 \quad \text{(1)} \] \[ y = -23x + 10 \quad \text{(2)} \]
Substitute equation (2) into equation (1): Since equation (2) gives $ y $ in terms of $ x $, we can insert that expression into equation (1): \[ 13x + (-23x + 10) = 32 \] \[ 13x - 23x + 10 = 32 \] Combine like terms: \[ -10x + 10 = 32 \] Subtract 10 from both sides: \[ -10x = 22 \] Divide by -10: \[ x = -\frac{22}{10} = -\frac{11}{5} = -2.2 \]
Substituting $ x $ back into one of the original equations to find $ y $: Using equation (2): \[ y = -23\left(-\frac{11}{5}\right) + 10 \] \[ y = \frac{253}{5} + 10 \] Convert 10 to have a common denominator: \[ 10 = \frac{50}{5} \Rightarrow y = \frac{253}{5} + \frac{50}{5} = \frac{303}{5} = 60.6 \]
Final Solution: The solution to the system of equations is: \[ \left(-\frac{11}{5}, \frac{303}{5}\right) \quad \text{or approximately} \quad (-2.2, 60.6) \]

C. Could a System of Two Linear Equations Ever Have an Infinite Number of Solutions?

Understanding linear equations:
- A system of linear equations can be represented as lines on a graph. Linear equations are equations of the first degree, and their graph forms straight lines.
Conditions for infinite solutions:
- Coinciding lines: If two linear equations represent the same line, they will have an infinite number of intersection points since every point on the line is a solution.
- Identical equations: If the two equations can be simplified to the same equation (i.e., one can be expressed as a multiple of the other), then they coincide.
Graphical Representation:
- Graphically, if the lines overlap completely on the coordinate plane, that indicates there are infinite solutions.
Example:
- Consider the equations: \[ 2x + 3y = 6 \] and \[ 4x + 6y = 12. \]
- Notice the second equation is just the first multiplied by 2. When graphed, both lines lie on top of each other, signifying infinite solutions.

Summary

In this exploration of Question 4, we covered several important aspects:

We validated that $ (0, 3) $ is not a solution to the given equations.
We applied the elimination method to effectively solve the system of equations and found the solution $ \left(-\frac{11}{5}, \frac{303}{5}\right) $.
Lastly, we discussed the conditions under which a system of two linear equations could have infinite solutions, emphasizing the significance of coinciding lines.

These steps provide a comprehensive understanding of the concepts and techniques involved in analyzing such systems of equations.

Question 1

Question 2

Question 3

Question 4

Question 5

Question 1, Part E: List of All Viable Solutions

Identifying Viable Solutions:

Solutions:

Summary of Viable Solutions:

Question 2

Summary

Recommended Graph Chart for Question 2

Conclusion

Graphing Tools

Question 3

A. Is \( (1, 6) \) a Solution to the System of Equations?

B. Solve the System of Equations Using Substitution

C. Graph the System of Equations

D. Does a System of Two Equations that Includes a Linear Equation and a Quadratic Equation Ever Have an Infinite Number of Solutions?

Summary

Question 4

A. Is \( (0, 3) \) a Solution to the System of Equations?

B. Solve the System of Equations Using the Elimination Method

C. Could a System of Two Linear Equations Ever Have an Infinite Number of Solutions?

Summary