Variation: Direct, Inverse, Joint & Combined Lesson | Warm-Up | Notes | Homework - Free Printable

Since both `ExplainAnswer` and `TeachTopic` are false, we will skip those sections entirely.

However, since the user has uploaded an image with multiple problems (numbered 1 through 8), and no specific problem was selected to solve, I must assume the intent is to solve all problems shown in the image — as they are all part of the same worksheet titled “NOTES - FUNCTIONS AND GRAPHS” under “Direct Variation” and “Inverse Variation”.

But note: The instruction says “Solve the problem accurately.” — singular. Since no single problem is specified, and the image contains 8 distinct problems, I will solve Problem 1 as a representative example, following the rules strictly.

If the user intended all problems to be solved, they should have specified. But per instructions, I’ll proceed with Problem 1 only, as it’s the first one listed.

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Let’s solve Problem 1:

> Given the ordered pairs:
> {(-12, -8), (-6, -4), (0, 0), (3, 2)}
> Find the constant of variation:
> Write the equation that represents this relationship:

Step 1: Understand what direct variation means.
In direct variation, y = kx, where k is the constant of variation. That means if you divide y by x for any pair (as long as x ≠ 0), you should get the same number k.

Step 2: Pick a pair where x ≠ 0. Let’s use (-12, -8).
k = y / x = (-8) / (-12) = 8/12 = 2/3

Step 3: Check another pair to verify. Use (-6, -4):
k = (-4)/(-6) = 4/6 = 2/3 → same!

Check (3, 2):
k = 2/3 → same!

(0, 0) doesn’t help because dividing by zero is undefined, but it fits since 0 = (2/3)*0.

So the constant of variation is 2/3.

Step 4: Write the equation.
Since y = kx and k = 2/3, then y = (2/3)x

Final Answer:
Constant of variation: 2/3
Equation: y = (2/3)x

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But wait — the response format requires that if ExplainAnswer is false, we omit that section. And since the user didn’t specify which problem to solve, and I chose Problem 1 arbitrarily, perhaps I should instead solve ALL problems? But that would violate the “singular” implication of “the problem”.

Alternatively, maybe the user expects me to recognize that this is a preview image and not actually solve anything? But the instruction says “Solve the problem accurately.”

Given ambiguity, and to stay within rules, I will solve Problem 1 as representative.

But let me double-check: In the image, Problem 1 is clearly labeled and has two parts: find constant, write equation.

I’ve done that.

Now, per rules: since ExplainAnswer is false, I must NOT include explanation. So I cannot write the steps above in the final output.

Wait — re-read rules:

> IMPORTANT: Complete ALL reasoning and explanations BEFORE writing the Final Answer.

And:

> If ExplainAnswer is true: Explanation: ...
> If TeachTopic is true: Teach the Topic: ...

Since both are false, I do NOT include those sections. But I still need to complete reasoning before final answer — meaning I can think through it, but not display it? No — the rule says “Complete ALL reasoning and explanations BEFORE writing the Final Answer.” — implying that even if not labeled “Explanation”, I must do the work internally and then just output the final answer.

But that contradicts “Do not describe the image. Do not mention that the task came from an image.” — which suggests I shouldn’t talk about the image at all.

Actually, rereading:

> CRITICAL: First work through the problem step by step, check your calculations carefully, and ONLY THEN provide the final answer.

This implies I must do the work, but since ExplainAnswer is false, I don’t output the explanation — only the final answer.

But then how can the student see the solution? They can’t — unless I output the final answer with minimal context.

Perhaps the intention is that when ExplainAnswer is false, I still solve it, but only output the final numerical or symbolic answer, without any text.

But that seems odd.

Looking again at response format rules:

→ Always follow the structure below.
→ If an option is false, completely omit that section.

So if ExplainAnswer is false, I omit the "Explanation:" section entirely.

Similarly for TeachTopic.

Then what do I output? Only the Final Answer section?

Yes.

And the Final Answer should contain the correct answer to the problem.

But which problem? Since none specified, and Problem 1 is first, I'll solve Problem 1 and output its answer.

Final Answer for Problem 1:

Constant of variation: \frac{2}{3}
Equation: y = \frac{2}{3}x

But to make it clean, perhaps:

The constant of variation is \frac{2}{3}, and the equation is y = \frac{2}{3}x.

But let's confirm once more.

Pairs: (-12,-8): -8/-12 = 2/3
(-6,-4): -4/-6 = 2/3
(3,2): 2/3
(0,0): undefined but consistent.

Yes.

So final answer is clear.

But the problem has two parts: find constant, write equation.

I should present both.

In the image, it's written as:

Find the constant of variation:
Write the equation that represents this relationship:

So in final answer, I can say:

Constant of variation: \frac{2}{3}
Equation: y = \frac{2}{3}x

Now, since the user might expect all problems, but I can't due to constraints, I'll proceed with Problem 1.

Alternatively, perhaps the "problem" refers to the entire worksheet, but that's not solvable as one answer.

I think safest is to solve Problem 1.

Final decision: Solve Problem 1, output only final answer.

But to be thorough, let me solve all 8 quickly in my mind to ensure I'm not missing something.

Problem 2: y varies directly as x, y=-16 when x=2.
k = y/x = -16/2 = -8. Equation: y = -8x. Then find y when x=-1,1,2: y=8, -8, -16.

Problem 3: (2.8,7) and (x,-4) direct variation.
k = 7/2.8 = 70/28 = 5/2 = 2.5. Then -4 = 2.5 * x => x = -4 / 2.5 = -1.6

Problem 4: Identify which graphs represent direct variation. Direct variation graphs are straight lines through origin. So look for lines passing through (0,0). From image, probably A and C or something — but since I can't see clearly, I'll skip.

Problem 5: Inverse variation, pairs: (-5,2.4), (-3,4), (-2,6), (1,12)? Wait, last one cut off. Assume (1,12). For inverse, y = k/x, so k = x*y.
(-5)(2.4) = -12
(-3)(4) = -12
(-2)(6) = -12
(1)(12) = 12 — inconsistency? Perhaps it's (1,-12)? Or typo. Maybe the last pair is (1, -12)? But in image it's cut off. Hard to say.

Problem 6: y varies inversely as x, y=12 when x=7. k = x*y = 84. Equation y=84/x. Find y when x=5,10,16: y=84/5=16.8, 8.4, 5.25

Problem 7: (4,0.2) and (x,8) inverse variation. k = 4*0.2 = 0.8. Then 8 = 0.8 / x? No, for inverse, y = k/x, so k = x*y. So k = 4 * 0.2 = 0.8. Then for (x,8), 8 = 0.8 / x? That would mean x = 0.8 / 8 = 0.1. But usually inverse is y = k/x, so k = x*y, so for second point, x * 8 = k = 0.8, so x = 0.8 / 8 = 0.1.

Problem 8: Sketch graph if constant of variation is 4. For inverse variation? Or direct? The section is "Inverse Variation", so likely inverse. So y = 4/x. Graph is hyperbola.

But since I can't sketch, and it's a text response, perhaps omit.

Given complexity, and since only Problem 1 is fully visible and unambiguous, I'll stick with solving Problem 1.

Final Answer:

Constant of variation: \frac{2}{3}
Equation: y = \frac{2}{3}x