Week 5 — Where Two Lines Meet: A Short Systems Report — Traditional alternative (no AI)

Instructors: publish this instead of the adaptive version if you prefer, and unpublish or delete the adaptive one.

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 week you learned what a solution of a system is, how to solve a system by substitution and by elimination, how to multiply an equation first, and how to read a leftover statement such as 0 = 4 or 0 = 0. In Week 4 you read one line at a time; this week you find where two lines meet. In this assignment you work four short problems, interpret each one, and explain to someone who has never studied algebra what a break-even point means. This week's lecture tutorial, quiz, discussion, and Model/Build Studio practise the same ideas from other angles.

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.

Show your arithmetic. A reader should be able to reproduce every number you report from the values above without guessing which values or steps you used.

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.

Deliverables

Submit a single systems report containing:

• the four answers, each with the arithmetic shown, the completed pair checked in both equations where there is one, and a brief interpretation of what that value shows and one use or limit of it;
• your message for the club treasurer, explaining what the 6 and the 90 in the break-even point mean, how to check it, and one honest limit; and
• a one-line restatement of the values you used, so your results are reproducible.

Keep it to roughly one page. Clear and correct beats long.

Choose how you show it. Present your report as written text, or as a recorded audio or video explanation of about three minutes with a transcript, or as a labelled sketch of the lines with a short note. Every form is graded by the same scoring criteria. 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.

How you'll be graded

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

Academic integrity and AI use

This version is designed to be completed without AI: do the computations, the interpretations, and the writing yourself, without a chatbot or other AI tool. You may use a calculator or a free graphing calculator such as Desmos to check your arithmetic and your lines. Submitting AI-generated work as your own, or copying another student's numbers, is an integrity violation under the course policy.

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