Question

Use the Two-variable Equations and Inequalities Portfolio worksheet to record your answers to the following questions. When you are finished, save your worksheet with your answers and submit it for a portfolio grade.

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 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.
Two-variable Equations and Inequalities Portfolio Worksheet

Use the rubric to check your work on the portfolio. 

Two-variable Equations and Inequalities Portfolio Rubric

When you are ready to submit your portfolio, be sure to include all of these items:

the completed Two-variable Equations and Inequalities Portfolio worksheet
your graph of the solution set of the two-variable equation for question 1C
your graph of the solution set of the two-variable inequality that includes all viable solutions to the problem as points on the graph for questions 2C and 2E
your graph of the system of equations with the solution(s) labeled for question 3C
your graph of your system of linear inequalities with the solution region clearly indicated for question 5C . Make the answer really short.

Answer 1

Below are very short answers.

Question 1
A. Let b = number of board books, s = number of soft-cover books. 1.50b + 0.50s = 20.
B. For (b,s)=(10,5): 1.5(10)+0.5(5)=15+2.5=17.5 ≠ 20, so no.
C. Graph: line 1.5b+0.5s=20 (b-intercept 40/3 ≈13.33, s-intercept 40).
D. Constraints: b ≥ 0, s ≥ 0, integers (whole books).
E. Integer solutions from 3b + s = 40 (multiply by 2): (b,s) = (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).
F. If total books =34: b + s =34 and 3b + s =40 ⇒ subtract ⇒ 2b=6 ⇒ b=3, s=31.

Question 2
A. Let c = children, a = adults. 150 + 10.5c + 5.5a ≤ 300 ⇒ 10.5c + 5.5a ≤ 150, with c ≥1, a ≥1 integers.
B. For (c,a)=(10,5): cost =150+105+27.5=282.5 ≤300, so yes.
C. Graph: half-plane 10.5c+5.5a ≤150 (boundary line 10.5c+5.5a=150).
D. Constraints: c ≥1, a ≥1, integers.
E. Viable solutions = integer (c,a) pairs with c≥1,a≥1 and 10.5c+5.5a ≤150 (e.g., a=1 ⇒ c ≤13; a=5 ⇒ c ≤14).
F. Recommendation (example): invite 10 children and 5 adults — within budget and reasonable supervision; other choices possible depending on supervision needs.

Question 3
System: −x + y = 5 and y = (x−3)^2 + 2.
A. (1,6)? Check: −1+6=5 and (1−3)^2+2=4+2=6 ⇒ yes.
B. Substitution: y = x+5. Set x+5 = (x−3)^2 +2 ⇒ x^2 −7x +6 =0 ⇒ (x−1)(x−6)=0 ⇒ x=1 or 6. Then y = x+5 ⇒ solutions (1,6) and (6,11).
C. Graph: line y=x+5 and parabola y=(x−3)^2+2 intersect at (1,6) and (6,11).
D. Infinite solutions? Only if the “quadratic” is actually the same as the line (i.e., quadratic degenerates to the same linear equation). Graphically that means the parabola and line coincide (which requires the quadratic to reduce to that line).

Question 4
System: 13x + y = 32 and y = −23x + 10.
A. (0,3)? 13·0+3=3 ≠32, so no.
B. Substitute y: 13x + (−23x+10)=32 ⇒ −10x +10=32 ⇒ −10x=22 ⇒ x = −11/5; y = −23(−11/5)+10 = 303/5. Solution (−11/5, 303/5).
C. Infinite solutions? Yes if the two equations are multiples (same line). Algebraically coefficients and constants proportional; graphically they coincide.

Question 5
A. Example system: y ≥ 2x + 1 and y < −x + 4.
B. To test a point (x0,y0), substitute into both inequalities; it must satisfy both. Example: (1,3): 3 ≥ 2(1)+1 → 3 ≥ 3 true; 3 < −1+4 → 3 < 3 false, so (1,3) is not in the solution.
C. Graph: shade region y ≥ 2x+1 and region below y < −x+4; solution is their intersection.
D. No solutions? Yes—if shaded regions don’t overlap (e.g., y > x+1 and y < x−1). On a graph this appears as two nonoverlapping regions with empty intersection.

(Do your graphs on graph paper and attach them where required.)