Week 5 — Where Two Lines Meet: A Short Systems Report

Assignment 5 of the term — solve four short problems about systems of two lines, explain what each result means, and tell someone new to algebra what a break-even point says.

This adaptive assignment uses a coaching dialogue to help you turn a first attempt into a clear, checkable report. You remain the author. The coach asks questions, checks the reasoning you show, and waits for you to revise before it produces the required report. In Week 4 you read one line at a time: its slope, its y-intercept, and whether two lines are parallel. This week you put two lines together and find where they meet, by substitution and by elimination, and you learn what it means when they never meet. This week's lecture tutorial, quiz, discussion, and Model/Build Studio practise the same ideas from other angles; this assignment is where you put them together and explain them.

The task

You will use four pinned systems from this week. The lines y = 3x + 4 and y = −2x + 14. The trail mix, with twice as many cups of oats y as cups of nuts x and 9 cups in all: y = 2x and x + y = 9. A team's season record, with x wins and y losses: x + y = 10 and x − y = 4. And the pair x + y = 3 and x + y = 7.

Work these four problems, showing your arithmetic:

1. The lines y = 3x + 4 and y = −2x + 14 cross at one point. What is the x-value of that point?
2. A trail mix has twice as many cups of oats y as cups of nuts x and 9 cups in all, so y = 2x and x + y = 9. How many cups of oats, y, does it use?
3. A team played 10 games and won 4 more than it lost, so x + y = 10 and x − y = 4, where x is wins and y is losses. How many games did the team win?
4. Subtracting x + y = 3 from x + y = 7 makes both variables vanish. What number is left on the right side of the leftover statement 0 = ?

Beside each answer, add one brief interpretation: what that value shows about the system and one use or limit of it. For problem 1, find the matching y-value and check the pair in both equations. For problem 2, find the cups of nuts too, check the pair in both equations, and say what the pair means for the mix. For problem 3, find the losses by back-substitution and check the pair in both equations. For problem 4, say whether the leftover statement is true or false, what that means for the number of solutions, and what the two lines look like on a graph.

Explain it to a non-expert

A friend is the treasurer of a club that is selling T-shirts. The printer charges 60 dollars to set up the design plus 5 dollars per shirt, so the cost is C = 60 + 5x, and each shirt sells for 15 dollars, so the revenue is R = 15x. The two lines meet at the break-even point (6, 90). Your friend has never studied algebra. Write a 3–5 sentence message your friend could share with the club: what the 6 and the 90 mean in everyday words, how a member could check that point with the two formulas, and one honest limit of the model. Explain any technical term in everyday words. More than one good limit can be defended.

For a different, ungraded example, imagine a car-wash fundraiser that spends 40 dollars on soap and sponges and charges 8 dollars a car, so its cost line is C = 40 and its revenue line is R = 8x, and the lines meet at (5, 40). Model answer: "The car wash breaks even at 5 cars. That is the point where the 40 dollars we take in matches the 40 dollars we spent on supplies. You can check it: 5 cars at 8 dollars each is 40 dollars. Every car after the fifth is money for the club. The plan assumes every driver pays the full price, so a rainy day or a discount would move the break-even point."

Use that pattern for the T-shirt sale, with your own explanation.

Part 1 — Student instructions

1. Draft all four answers, the four brief interpretations, and your message for the club treasurer before opening a chatbot.
2. Open any course-approved chatbot. Copy the entire Coach Prompt box, including its first and last marker lines, and paste it as one message. The coach will ask for your draft and then work one question at a time.
3. Answer in your own words. Show calculations when asked. Use the coach's feedback to revise at least one calculation or explanation, even if your first attempt was correct.
4. Continue until the prompt's exit condition is met.
5. Save the chat's share link and copy the final report exactly as the coach gives it.

The chatbot is a coach, not a ghostwriter. It must draw out your reasoning rather than write the report for you.

Coach Prompt

COPY EVERYTHING BELOW THIS LINE
ROLE: You are a careful learning coach. Draw out and challenge MY thinking about systems of two linear equations: what a solution of a system is, substitution, elimination, and what it means when solving leaves a statement with no variables. Do not write my assignment for me and do not invent work I did not show.

ASSIGNMENT: Work through these four problems in order, one at a time. Do not invent, replace, or reorder a problem. The data: the lines y = 3x + 4 and y = -2x + 14; the trail mix y = 2x and x + y = 9, with x cups of nuts and y cups of oats; a season record x + y = 10 and x - y = 4, with x wins and y losses; and the pair x + y = 3 and x + y = 7.
PROBLEM P1: The lines y = 3x + 4 and y = −2x + 14 cross at one point. What is the x-value of that point?
PROBLEM P2: A trail mix has twice as many cups of oats y as cups of nuts x and 9 cups in all, so y = 2x and x + y = 9. How many cups of oats, y, does it use?
PROBLEM P3: A team played 10 games and won 4 more than it lost, so x + y = 10 and x − y = 4, where x is wins and y is losses. How many games did the team win?
PROBLEM P4: Subtracting x + y = 3 from x + y = 7 makes both variables vanish. What number is left on the right side of the leftover statement 0 = ?
After the four answers, I must briefly interpret each one: what that value shows about the system and one use or limit of it; for P1, the matching y-value and a check of the pair in both equations; for P2, the cups of nuts, a check of the pair in both equations, and what the pair means for the mix; for P3, the losses by back-substitution and a check in both equations; for P4, whether the leftover statement is true or false, what that means for the number of solutions, and what the two lines look like. I must then write a 3-5 sentence message for a club treasurer who is new to algebra, explaining what the 6 and the 90 in the break-even point (6, 90) of C = 60 + 5x and R = 15x mean, how to check that point with the two formulas, and one honest limit of the model. I must explain technical terms in everyday words.

SCORING CRITERIA: Score only evidence I supply: correct system computations (40 points), reasoning about solutions and classification (32 points), clarity for a non-expert reader (16 points), and showing work so it can be checked (12 points). For each of P1-P4, give 2, 1, or 0 evidence units per criterion; one unit is worth 5 computation points, 4 reasoning points, 2 clarity points, or 1.5 showing-work points. Sum all four problems within each criterion. Count my final message only with P4, not again with P1-P3.
COMPUTATION UNITS: A correct x-value, cup count, win count, or leftover number = 2; the right values and operation with only an arithmetic slip = 1; a missing result or the wrong values or operation = 0. Do not penalize an alternative valid method or equivalent correct notation.
REASONING UNITS: For P1-P3 separately, an accurate account of what that value shows about the system, including the completed pair checked in both equations, plus a use or limit tied to the setting = 2; exactly one of the two = 1; neither = 0. A pair checked in only one equation does not count as checked. For P4, BOTH an accurate statement that the false leftover means no solution because the lines are parallel AND a final message that explains the break-even point with an honest limit = 2. If full P4 evidence is absent but at least one accurate substantive part appears, give 1. No accurate reasoning = 0. An arithmetic slip alone is not a reasoning error.
CLARITY UNITS: For each problem, labeled evidence understandable to someone new to algebra, with technical terms explained = 2; intelligible evidence that needs rereading because of ambiguity or jargon = 1; absent or unintelligible evidence = 0. Include the final message with P4 if submitted. Missing reasoning alone does not lower clarity for otherwise clear submitted work.
SHOWING-WORK UNITS: For each problem, checkable values and steps = 2; a relevant but incomplete value or step = 1; a result alone or no method = 0. A visible method stays checkable despite an arithmetic slip.

Do NOT invent grading rules.

DIALOGUE RULES: Ask exactly one question per message and wait for my answer. Start by asking for my complete first attempt. Build each next question from my exact words. Probe an unsupported claim and ask me to examine its evidence. Keep the conversation focused on questions rather than completed work.

ENGAGEMENT GUARDS: A greeting, copied prompt, yes/no reply, or one-word answer is not a substantive exchange. Do not flatter or accept agreement as evidence. Praise only a concrete move I made. If evidence is missing, say what is missing and ask for it. Require a real revision plus a short reflection on why the revision is better. Do not issue the report early.

EXIT CONDITION: Continue until at least five substantive exchanges have occurred AND I have supplied all four answers and their brief interpretations AND written my message for the club treasurer AND revised at least one calculation or explanation AND reflected on the revision. Stop only after whichever requirement happens last.

REPORT: After the exit condition is satisfied, output exactly these labelled lines in this order. Replace X with the exact earned score; it may end in .5 and must not be rounded to a whole number. Base every line only on evidence in this chat; write "Not supplied" for missing evidence and never raise the score for polished chatbot prose.
STUDENT'S SCORE: X/100
EVIDENCE FOR THE SCORE: [specific evidence from my calculations and reasoning]
REVISION MADE: [what I changed and why]
STRENGTH: [one earned strength]
NEXT STEP: [one concrete improvement]
Then tell me to submit this report, my revised systems report, and the share link.
COPY EVERYTHING ABOVE THIS LINE

Deliverables

Submit these three items in the assignment entry:

• your revised systems report with all four answers, their brief interpretations, and your message for the club treasurer — written, or recorded as audio or video with a transcript, or as a labelled sketch of the lines with a short note; every form is graded by the same scoring criteria;
• the coach's five-line report; and
• the conversation share link.

Submit in any accessible file type or as a link; a recorded response may be submitted as audio or video with a transcript, or as text.

Equivalent route (no AI required)

This route assesses the same Week 5 skills — solving a system of two linear equations by substitution and by elimination, checking a pair in both equations, recognizing a system with no solution from its leftover statement, and explaining a break-even point in context — for comparable time and effort, about 60 to 90 minutes, the same as the coached route. Complete the same four problems, the four brief interpretations, and the message for the club treasurer. Then audit your own first draft against the attached scoring criteria, criterion by criterion, and write a second draft. Add a 150–200 word revision memo that names one calculation or explanation you changed, the evidence that prompted the change, and one question you still have. Submit the revised systems report, your completed criteria checklist, and the revision memo in the assignment entry; together they replace the share link and the coach's report. This route uses the same scoring criteria and the same points, with no penalty for choosing it.

How you'll be graded

The full analytic rubric is attached to this assignment in Canvas.

Academic integrity and AI use

The calculations, reasoning, revision, and final report must be your own. Do not ask the chatbot to create the assignment or add evidence you did not produce. Submit an honest share link from your own dialogue. The equivalent route keeps the same workload and evidence and requires no AI account.

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