x + y = 10xy = 21The equations above represent relationships between x and y. What is a possible value...

Part 1: Brief Solution

Concepts tested: Graph intersections, systems of equations, quadratic solving
Primary process skills: Translate intersection condition, Infer simultaneous satisfaction, Consider all cases

Essential steps:

• Recognize that intersection means both equations are satisfied: \(\mathrm{y = 76}\) and \(\mathrm{y = x^2 - 5}\)
• Set expressions equal: \(\mathrm{76 = x^2 - 5}\)
• Solve: \(\mathrm{x^2 = 81}\), so \(\mathrm{x = \pm 9}\)
• Verify: Both \(\mathrm{x = 9}\) and \(\mathrm{x = -9}\) work; answer choice B gives \(\mathrm{x = -9}\)

Answer: B) -9

Part 2: Top 3 Faltering Points

Top 3 Faltering Points:

1. Intersection Misconception - Phase: Devising Approach → Choice D (76)
   • Process skill failure: Translate
   • Students confuse the y-coordinate of intersection (76) with the x-coordinate being asked for.
2. Missing Negative Solution - Phase: Executing Approach → Various choices except B
   • Process skill failure: Consider all cases
   • Students find \(\mathrm{x = 9}\) from \(\mathrm{x^2 = 81}\) but forget that \(\mathrm{x = -9}\) is also valid, missing the available answer choice.
3. Wrong Equation Setup - Phase: Devising Approach → Choice C (5)
   • Process skill failure: Infer
   • Students incorrectly set \(\mathrm{x^2 - 5 = 0}\) instead of recognizing that intersection requires \(\mathrm{x^2 - 5 = 76}\).

Part 3: Detailed Solution

When two graphs intersect, they share the same point (x, y). This means that at the intersection, both equations must give the same y-value for the same x-value.

Process Skill: TRANSLATE - Let me convert the intersection condition into mathematical language. Since both graphs pass through the same point, we have:

• Point satisfies first equation: \(\mathrm{y = 76}\)
• Point satisfies second equation: \(\mathrm{y = x^2 - 5}\)
• At intersection: both expressions equal the same y-value

Process Skill: INFER - Since both expressions equal y, I can set them equal to each other:
\(\mathrm{76 = x^2 - 5}\)

Now I'll solve this equation step by step:
\(\mathrm{x^2 - 5 = 76}\)

Add 5 to both sides:
\(\mathrm{x^2 = 76 + 5}\)
\(\mathrm{x^2 = 81}\)

Process Skill: SIMPLIFY - I need to solve \(\mathrm{x^2 = 81}\). Taking the square root of both sides:
\(\mathrm{x = \pm\sqrt{81}}\)
\(\mathrm{x = \pm 9}\)

This gives me two solutions: \(\mathrm{x = 9}\) and \(\mathrm{x = -9}\).

Process Skill: CONSIDER ALL CASES - It's crucial to remember that when solving \(\mathrm{x^2 = 81}\), there are always two solutions (unless we're told the domain restricts us to positive values only). Both \(\mathrm{x = 9}\) and \(\mathrm{x = -9}\) are mathematically valid.

Let me verify both solutions:

• For \(\mathrm{x = 9}\): \(\mathrm{y = 9^2 - 5 = 81 - 5 = 76}\) ✓
• For \(\mathrm{x = -9}\): \(\mathrm{y = (-9)^2 - 5 = 81 - 5 = 76}\) ✓

Both solutions work! Looking at the answer choices, I can see that B) -9 is one of the valid solutions.

Part 4: Detailed Faltering Points Analysis

Errors while devising the approach:

Intersection Misconception (Process skill: Translate): Many students misunderstand what "intersection" means algebraically. They might think they need to solve each equation separately, or they confuse which coordinate is being asked for. Some students see \(\mathrm{y = 76}\) and immediately think the answer is 76, not realizing that 76 is the y-coordinate and we need the x-coordinate.

Wrong Equation Setup (Process skill: Infer): Students might incorrectly set up the equation as \(\mathrm{x^2 - 5 = 0}\), thinking they need to find where the parabola crosses the x-axis, rather than where it intersects the horizontal line \(\mathrm{y = 76}\). This leads to \(\mathrm{x^2 = 5}\), giving \(\mathrm{x = \pm\sqrt{5} \approx \pm 2.24}\), which might round to answer choice C (5).

Graphical Misinterpretation (Process skill: Translate): Some students might try to visualize this graphically but misinterpret what the intersection means, perhaps thinking they need to find where \(\mathrm{y = 76}\) intersects \(\mathrm{y = x^2 - 5}\) in some other way.

Errors while executing the approach:

Algebraic Computation Errors (Computational error): Students might make arithmetic mistakes such as:

• \(\mathrm{76 + 5 = 80}\) instead of 81
• Incorrectly calculating \(\mathrm{(-9)^2}\) as -81 instead of +81
• Forgetting to add 5 when moving it to the other side

Missing Negative Solution (Process skill: Consider all cases): The most common execution error is finding \(\mathrm{x^2 = 81}\), correctly determining \(\mathrm{x = 9}\), but forgetting that \(\mathrm{x = -9}\) is also a valid solution. Since only -9 appears in the answer choices, students who forget the negative solution won't find their answer among the choices.

Square Root Errors (Computational error): Students might incorrectly calculate \(\mathrm{\sqrt{81}}\), though this is less likely since 81 is a common perfect square.

Errors while selecting the answer:

Solution-Choice Mismatch: Students who find \(\mathrm{x = 9}\) but don't see it among the choices might panic and select a different answer rather than checking for the negative solution.

Y-coordinate Confusion: Students might select choice D (76) thinking this is the answer, confusing the y-coordinate of the intersection with the x-coordinate being asked for.

Approximation Errors: Students who made computational mistakes earlier might try to match their incorrect result to the closest answer choice, potentially selecting choice A (\(\mathrm{76/5 = 15.2}\)) if they got something close to 15.