\(\mathrm{y - 1 = 2(x - 3)}\)\(\mathrm{y - 1 = 2(x - 3)^2}\)Which of the following ordered pairs \(\mathrm{(x, y)}\)...

1. TRANSLATE the problem information

Given system:

• \(\mathrm{y - 1 = 2(x - 3)}\) [linear equation]
• \(\mathrm{y - 1 = 2(x - 3)^2}\) [quadratic equation]

We need to find ordered pairs \(\mathrm{(x, y)}\) that satisfy both equations.

2. INFER the solution strategy

Since both equations equal \(\mathrm{y - 1}\), we can set the right sides equal to each other:

\(\mathrm{2(x - 3) = 2(x - 3)^2}\)

This eliminates y and gives us an equation in x only.

3. SIMPLIFY the equation

Divide both sides by 2:

\(\mathrm{(x - 3) = (x - 3)^2}\)

Let \(\mathrm{u = x - 3}\) to make this clearer:

\(\mathrm{u = u^2}\)

Rearrange: \(\mathrm{u^2 - u = 0}\)

Factor: \(\mathrm{u(u - 1) = 0}\)

4. CONSIDER ALL CASES for the solutions

From \(\mathrm{u(u - 1) = 0}\), we get:

• \(\mathrm{u = 0}\), which means \(\mathrm{x - 3 = 0}\), so \(\mathrm{x = 3}\)
• \(\mathrm{u = 1}\), which means \(\mathrm{x - 3 = 1}\), so \(\mathrm{x = 4}\)

5. Find corresponding y-values

For \(\mathrm{x = 3}\):

\(\mathrm{y - 1 = 2(3 - 3) = 0}\)

\(\mathrm{y = 1}\) → \(\mathrm{(3, 1)}\)

For \(\mathrm{x = 4}\):

\(\mathrm{y - 1 = 2(4 - 3) = 2}\)

\(\mathrm{y = 3}\) → \(\mathrm{(4, 3)}\)

6. APPLY CONSTRAINTS from answer choices

The system has solutions \(\mathrm{(3, 1)}\) and \(\mathrm{(4, 3)}\). Among the given choices, only \(\mathrm{(3, 1)}\) appears.

Answer: B

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Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize that they can set the two expressions equal since both equal \(\mathrm{y - 1}\). Instead, they try to solve each equation separately or attempt substitution in a more complicated way.

This leads to confusion about how to combine the equations systematically and often results in guessing among the answer choices.

Second Most Common Error:

Inadequate SIMPLIFY execution: Students make algebraic errors when working with \(\mathrm{(x - 3) = (x - 3)^2}\), either by not properly factoring \(\mathrm{u^2 - u = 0}\) or by making calculation mistakes when finding the x and y values.

This may lead them to select Choice A (2, -1) or Choice C (3, 3) based on incorrect calculations.

The Bottom Line:

This problem tests whether students can recognize the elegant approach of setting expressions equal when they equal the same variable, rather than getting bogged down in more complex substitution methods. The key insight is realizing the structural relationship between the equations.