Solved Algebra 2 Assignment Identify the domain and range of ... - Free Printable

Let’s solve Exercise 25 step by step.

We are given the system:

> y = (1/2)x² – 5x + 21/2
> y = –(1/2)x + 13/2

We need to find which ordered pairs from the options are solutions — meaning, they satisfy both equations.

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Step 1: Set the two expressions for y equal to each other

Since both equal y, we can set them equal:

(1/2)x² – 5x + 21/2 = –(1/2)x + 13/2

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Step 2: Eliminate fractions by multiplying everything by 2

Multiply every term by 2:

→ x² – 10x + 21 = –x + 13

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Step 3: Move all terms to one side

Add x and subtract 13 from both sides:

→ x² – 10x + x + 21 – 13 = 0
→ x² – 9x + 8 = 0

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Step 4: Factor the quadratic

Look for two numbers that multiply to 8 and add to –9 → those are –1 and –8.

So:

→ (x – 1)(x – 8) = 0

Thus, x = 1 or x = 8

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Step 5: Find corresponding y-values using the simpler equation

Use y = –(1/2)x + 13/2

- If x = 1:
y = –(1/2)(1) + 13/2 = –0.5 + 6.5 = 6
→ Point: (1, 6)

- If x = 8:
y = –(1/2)(8) + 13/2 = –4 + 6.5 = 2.5
→ Point: (8, 2.5)

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Step 6: Check against answer choices

Options:

A) (1, 6) ✔
B) (3, 0)
C) (8, 2.5) ✔
D) (7, 0)

Both A and C match our solutions.

But wait — the question says: “Which ordered pairs are solutions?” and gives single-letter options. That suggests it might be multiple choice with only one correct letter — but looking again, the format shows A, B, C, D as separate options, and possibly more than one could be correct? But in standard multiple-choice like this, usually only one is selected.

Wait — let me re-read the problem:

> 25. USING EQUATIONS Which ordered pairs are solutions of the nonlinear system?

And then lists A, B, C, D as individual points.

In many textbooks, when they say “which ordered pairs” and list options like this, sometimes you’re supposed to pick all that apply, but here the answer format seems to expect a single letter.

But look — both (1,6) and (8,2.5) are mathematically correct.

However, checking the original image layout — it's likely that this is a multiple-select question, but since the user didn’t specify, and the instruction says “provide only the final correct answer”, and the options are labeled A, B, C, D individually...

Wait — perhaps I made a mistake? Let me verify both points in both equations.

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Verify (1, 6):

First equation: y = (1/2)(1)² – 5(1) + 21/2 = 0.5 – 5 + 10.5 = (0.5 + 10.5) – 5 = 11 – 5 = 6 ✔
Second equation: y = –(1/2)(1) + 13/2 = –0.5 + 6.5 = 6 ✔

Verify (8, 2.5):

First equation: y = (1/2)(64) – 5(8) + 21/2 = 32 – 40 + 10.5 = (32 + 10.5) – 40 = 42.5 – 40 = 2.5 ✔
Second equation: y = –(1/2)(8) + 13/2 = –4 + 6.5 = 2.5 ✔

So both A and C are correct.

But the way the question is formatted — with circles around A, B, C, D — suggests it may be expecting us to choose one option. However, in some systems, you can select multiple. Since the problem says “ordered pairs” (plural), and both A and C work, perhaps the intended answer includes both.

But looking back at the image description — actually, in the original worksheet, Exercise 25 has four options labeled A, B, C, D, and typically in such formats, if multiple are correct, it would say “select all that apply”. Since it doesn’t, maybe there’s an error?

Wait — let me check option B: (3,0)

Plug into second equation: y = –(1/2)(3) + 13/2 = –1.5 + 6.5 = 5 ≠ 0 ✘

Option D: (7,0)

y = –(1/2)(7) + 13/2 = –3.5 + 6.5 = 3 ≠ 0 ✘

So only A and C work.

But since the problem likely expects a single answer, and both A and C are listed separately, perhaps the question allows selecting multiple? Or maybe I misread.

Wait — looking again at the user’s upload: the text says “Which ordered pairs are solutions” — plural — so probably more than one.

But in the response format, I must give a Final Answer. And since both A and C are correct, and the problem didn’t specify “choose one”, I should report both.

However, in standardized testing, sometimes they design it so only one is correct — but here clearly two are.

Perhaps the answer is to list both letters.

But let me see the exact wording again:

> 25. USING EQUATIONS Which ordered pairs are solutions of the nonlinear system?

Then lists:

A (1,6)
B (3,0)
C (8,2.5)
D (7,0)

In many algebra books, when they present it this way, and ask “which ordered pairs”, they expect you to identify all correct ones. So the correct answers are A and C.

But since the Final Answer section requires a single entry, and the options are labeled with letters, I think the expected format is to list the correct letters.

Alternatively, perhaps the problem is designed so that only one is correct — but our math shows two.

Wait — let me double-check the first equation at x=8:

y = (1/2)(8)^2 - 5*8 + 21/2 = (1/2)*64 = 32; -5*8=-40; 21/2=10.5 → 32 - 40 = -8; -8 + 10.5 = 2.5 ✔

Yes.

At x=1: 0.5 - 5 + 10.5 = 6 ✔

So both are valid.

Given that, and since the problem says "ordered pairs" (plural), I will conclude that both A and C are correct.

But how to write the Final Answer? The instruction says: “Provide only the final correct answer”

Looking at similar problems, sometimes they expect you to list the letters. Perhaps the answer is A and C.

But in the context of the worksheet, maybe it's a multiple-answer question. Since the user didn't specify, and to follow instructions strictly, I'll provide both correct options.

However, reviewing the initial request: the user said "Solve the problem accurately." and the problem is to identify which ordered pairs are solutions.

So the accurate answer is that (1,6) and (8,2.5) are solutions, which correspond to options A and C.

Therefore, the final answer should include both.

But let me see if there's any trick. Is 2.5 written as 5/2? In the option it's written as (8, 2.5), which is fine.

Perhaps the book expects fractional form, but 2.5 is acceptable.

I think it's safe to go with A and C.

But since the Final Answer section might expect a single string, I'll write "A and C".

Alternatively, in some systems, they use commas.

To be precise, I'll state:

Final Answer: A, C

But let me confirm once more with substitution into the first equation for (8,2.5):

y = 0.5*(64) - 40 + 10.5 = 32 - 40 + 10.5 = (32 + 10.5) - 40 = 42.5 - 40 = 2.5 ✓

And for (1,6): 0.5 - 5 + 10.5 = 6 ✓

Perfect.

So the solutions are A and C.

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Final Answer: A, C