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

Answer 1

Below are complete answers you can record on your Two-variable Equations and Inequalities Portfolio worksheet. Where the worksheet asks for graphs, I describe the lines, intercepts, and shading so you can draw them on graph paper or insert photos/scans into your submission.

Question 1
A. Define variables and write equation
- Let b = number of board books, s = number of soft-cover books.
- Board books cost $1.50, soft covers $0.50, total spent $20.
- Equation: 1.50b + 0.50s = 20.

B. Check whether (b, s) = (10, 5) is possible
- Compute cost: 1.50(10) + 0.50(5) = 15 + 2.5 = 17.5 ≠ 20.
- Conclusion: No — 10 board books and 5 soft cover books would cost $17.50, not $20.

C. Graph the solution set (draw on graph paper)
- Solve for s: 0.5s = 20 − 1.5b → s = 40 − 3b.
- Intercepts: b = 0 → s = 40 (point (0,40)); s = 0 → b = 40/3 ≈ 13.33 (point (13.33,0)).
- On graph paper draw the straight line through (0,40) and (13.33,0). (At this stage you graph the full line — do not yet limit to integer or nonnegative values.)

D. Constraints of the situation
- b ≥ 0 and s ≥ 0 (cannot buy negative books).
- b and s must be whole numbers (you can’t buy fractional books).
- In practice both are integers (0, 1, 2, ...).
- Also limited by s being integer from equation and b ≤ 13 from intercept if s≥0.

E. List all viable solutions (integer nonnegative solutions)
- Multiply the equation by 2 to eliminate decimals: 3b + s = 40 → s = 40 − 3b.
- For integer b ≥ 0 with s ≥ 0, b can be 0,1,...,13. The pairs:
(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).
- These are all the viable (integer, nonnegative) solutions.

F. If Santiago bought 34 books, how many of each type?
- Additional equation: b + s = 34.
- Solve system: 3b + s = 40 and b + s = 34.
Subtract: (3b+s) − (b+s) = 40 − 34 ⇒ 2b = 6 ⇒ b = 3.
Then s = 34 − 3 = 31.
- Conclusion: If he bought 34 books for $20, he bought 3 board books and 31 soft cover books.

Question 2
A. Define variables and write inequality
- Let C = number of children, A = number of adults.
- Cost = 150 + 10.50C + 5.50A. Budget maximum $300.
- Inequality: 150 + 10.50C + 5.50A ≤ 300.
(Also C ≥ 1 and A ≥ 1 are required by the situation.)

B. Check whether 10 children and 5 adults is possible
- Cost = 150 + 10.50(10) + 5.50(5) = 150 + 105 + 27.5 = 282.50 ≤ 300.
- Conclusion: Yes — 10 children and 5 adults fits within the $300 budget.

C. Graph the solution set (draw on graph paper)
- Rearrange: 10.5C + 5.5A ≤ 150 → A ≤ (150 − 10.5C)/5.5.
- Intercepts for the boundary line 10.5C + 5.5A = 150:
- C = 0 → A = 150/5.5 ≈ 27.27 (point (0,27.27)).
- A = 0 → C = 150/10.5 ≈ 14.286 (point (14.286,0)).
- Graph the boundary line through those intercepts. Shade the region below and/or on the line (because ≤).
- At this stage don’t yet limit to integers or to C,A ≥ 1 (except label axes).

D. Constraints of the situation
- C ≥ 1 and A ≥ 1 (at least one child and one adult).
- C and A must be whole numbers (counts of people).
- Also budget constraint already given: 150 + 10.5C + 5.5A ≤ 300.

E. Show all viable solutions as points
- Multiply inequality by 2 to avoid decimals: 300 + 21C + 11A ≤ 600 → 21C + 11A ≤ 300.
- With C and A integers and ≥1, find integer pairs (C, A) satisfying 21C + 11A ≤ 300.
- For each integer C = 1..13, the maximum integer A is:
C=1 → A ≤ 25
C=2 → A ≤ 23
C=3 → A ≤ 21
C=4 → A ≤ 19
C=5 → A ≤ 17
C=6 → A ≤ 15
C=7 → A ≤ 13
C=8 → A ≤ 12
C=9 → A ≤ 10
C=10 → A ≤ 8
C=11 → A ≤ 6
C=12 → A ≤ 4
C=13 → A ≤ 2
(C = 14 gives A ≤ 0, so not allowed since A ≥ 1.)
- All viable integer solutions are the sets of points (C,A) with C from 1 to 13 and A from 1 up to the corresponding max above. Plot each admissible integer pair as a point inside the shaded region (and respecting C,A ≥1).

F. Recommendation to Meha (one reasonable suggestion and justification)
- Recommendation: Invite about 12 children and 3 adults.
- Cost check: 150 + 10.5(12) + 5.5(3) = 150 + 126 + 16.5 = 292.50 ≤ 300.
- Justification: This uses the budget well while keeping an adult-to-child ratio of 1:4 (3 adults for 12 children), which is a reasonable supervision ratio for a children’s science center party. Alternatively, 10 children and 5 adults (cost $282.50) improves supervision (1:2) but invites fewer kids. Choose based on how much supervision the aunt wants vs. how many children to include.

Question 3
System (interpreting the text):
- Equations are: −x + y = 5 and y = (x − 3)^2 + 2.
- Equivalently y = x + 5 and y = (x − 3)^2 + 2.

A. Is (1,6) a solution?
- Check first: y = x + 5 → 6 = 1 + 5 = 6 ✔.
- Check second: y = (1 − 3)^2 + 2 = 4 + 2 = 6 ✔.
- Conclusion: Yes, (1,6) satisfies both equations.

B. Solve by substitution
- Set x + 5 = (x − 3)^2 + 2.
- Expand and simplify: x + 5 = x^2 − 6x + 9 + 2 → x + 5 = x^2 − 6x + 11.
- Move all to one side: 0 = x^2 − 6x + 11 − x − 5 = x^2 − 7x + 6.
- Factor: x^2 − 7x + 6 = (x − 1)(x − 6) = 0 → x = 1 or x = 6.
- Find y from y = x + 5: if x = 1 → y = 6; if x = 6 → y = 11.
- Solutions: (1, 6) and (6, 11).

C. Graph (draw on graph paper)
- Line: y = x + 5 (straight line slope 1, y-intercept 5).
- Parabola: y = (x − 3)^2 + 2, vertex at (3, 2), opens upward.
- Parabola intersects the line at (1,6) and (6,11). Label these points on your graph.

D. Could a linear equation and a quadratic equation have infinitely many solutions?
- No, not if the second equation is truly quadratic (degree 2). A quadratic and a linear equation generally intersect in at most 2 points (solutions of a quadratic equation in x).
- You could only have infinitely many solutions if the "quadratic" actually reduced to the same linear equation (i.e., its x^2 coefficient is 0 so both equations are the same line). From the graph, infinite solutions would appear only if the parabola degenerates into the same line (not a true parabola), so in the normal quadratic vs. linear case you cannot have infinitely many solutions.

Question 4
System (interpreting the text):
- Equations are: 13x + y = 32 and y = −23x + 10.

A. Is (0,3) a solution?
- Plug into first: 13(0) + 3 = 3 ≠ 32 ⇒ fails.
- Conclusion: No, (0,3) is not a solution.

B. Solve by elimination
- Rewrite second as 23x + y = 10 (moving terms to left).
- System:
13x + y = 32
23x + y = 10
- Subtract the second equation from the first:
(13x + y) − (23x + y) = 32 − 10 ⇒ −10x = 22 ⇒ x = −11/5.
- Substitute into y = −23x + 10:
y = −23(−11/5) + 10 = 253/5 + 50/5 = 303/5.
- Solution: (x, y) = (−11/5, 303/5). (You can also report decimal approximations x = −2.2, y = 60.6.)

C. Could two linear equations have infinitely many solutions?
- Yes — if the two equations represent the same line (one equation is a constant multiple of the other).
- How to tell from equations: If you can multiply one entire equation by a constant k (nonzero) to get the other exactly, they are the same line and have infinitely many shared points.
- How to tell from the graph: You would see the two lines coincide (overlap) exactly.
- (If the lines are parallel but not the same, there are no solutions; if they are distinct with different slopes, there is exactly one solution.)

Question 5
A. Write your own system of linear inequalities
- Example system:
(1) y ≥ 2x + 1
(2) y < −x + 4

B. How to determine whether a given point is a solution (description + example)
- To test a point (x0, y0), substitute x0 and y0 into both inequalities.
- If both inequalities are true after substitution, the point is a solution.
- If either inequality is false, the point is not a solution.
- Example: test (1,3):
- Check (1): 3 ≥ 2(1) + 1 → 3 ≥ 3 → true.
- Check (2): 3 < −1 + 4 → 3 < 3 → false (strict <).
- Conclusion: (1,3) is not a solution because it fails inequality (2).

C. Graph the solution set (draw on graph paper)
- Boundary 1: y = 2x + 1 (solid line because ≥). Shade the region above and on this line.
- Boundary 2: y = −x + 4 (dashed line because <). Shade the region below this line (not including the line).
- The solution region is the intersection of the two shaded half-planes: the area above y = 2x + 1 and strictly below y = −x + 4.
- Intersection point of the boundaries occurs at 2x + 1 = −x + 4 → 3x = 3 → x = 1 → y = 3. Because the second inequality is strict, the point (1,3) is not included even though it's the crossing of the lines. Mark the region clearly.

D. Could a system of linear inequalities have no solutions? What would that look like?
- Yes. If the solution regions of the inequalities do not overlap, the system has no solution.
- Graphically it looks like two non-overlapping half-planes (no common shaded area). For example:
- y > x + 1 and y < x − 1 simultaneously is impossible (no y can be strictly greater than x+1 and strictly less than x−1 at the same time), so the overlap is empty. On the graph you would see two parallel bands with a gap between them and no points in common.

Final checklist for your portfolio submission
- Include the completed worksheet with the equations, checks, lists, and explanations above.
- Include the graphs for:
- Q1C: the line 1.5b + 0.5s = 20 with intercepts (0,40) and (13.33,0).
- Q2C and Q2E: boundary 10.5C + 5.5A = 150 and the shaded budget region; plot all viable integer (C,A) points and mark constraints C≥1, A≥1.
- Q3C: parabola y = (x − 3)^2 + 2 and line y = x + 5 with intersection points (1,6) and (6,11) labeled.
- Q5C: the two boundary lines and the intersection shading region (clearly indicate included/excluded boundaries).
- Use the rubric on your worksheet to verify completeness before submitting.

If you want, I can prepare printable images of the graphs (line + parabola, shaded inequality regions, point lists) that you can insert into the portfolio — tell me which graphs you’d like as images.