Question:y = 2x + 10y = x^2 + 8x + 15A solution to the given system of equations is \(\mathrm{(x,y)}\)....

1. TRANSLATE the problem information

• Given system of equations:
  • \(\mathrm{y = 2x + 10}\) (linear equation)
  • \(\mathrm{y = x^2 + 8x + 15}\) (quadratic equation)

• We need to find the greatest possible value of x where these equations intersect

2. INFER the solution strategy

• Since both expressions equal y, we can set the right sides equal to each other
• This eliminates y and gives us an equation with only x
• We can solve for x, then determine which value is greatest

3. Set up the equation by INFERring the substitution

\(\mathrm{2x + 10 = x^2 + 8x + 15}\)

4. SIMPLIFY by rearranging to standard quadratic form

Move all terms to one side:
\(\mathrm{0 = x^2 + 8x + 15 - 2x - 10}\)
\(\mathrm{0 = x^2 + 6x + 5}\)

5. SIMPLIFY by factoring the quadratic

Look for two numbers that multiply to 5 and add to 6: that's 1 and 5
\(\mathrm{0 = (x + 1)(x + 5)}\)

6. CONSIDER ALL CASES by finding both solutions

Using zero product property:
\(\mathrm{x + 1 = 0}\) → \(\mathrm{x = -1}\)
\(\mathrm{x + 5 = 0}\) → \(\mathrm{x = -5}\)

7. APPLY CONSTRAINTS to select the greatest value

Between -1 and -5, the greatest value is -1.

Answer: -1

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak CONSIDER ALL CASES skill: Students find one solution but don't find both solutions to the quadratic equation.

Some students might factor correctly to get \(\mathrm{(x + 1)(x + 5) = 0}\), but then only solve \(\mathrm{x + 1 = 0}\) to get \(\mathrm{x = -1}\), missing that \(\mathrm{x + 5 = 0}\) gives \(\mathrm{x = -5}\). While this happens to give the right answer in this case, it shows incomplete understanding. Other students might only find \(\mathrm{x = -5}\) and miss \(\mathrm{x = -1}\).

This incomplete solution process can lead to selecting Choice A (-5) if they only found the smaller solution.

Second Most Common Error:

Poor SIMPLIFY execution: Students make algebraic errors when rearranging to standard form or factoring.

Common mistakes include sign errors when moving terms (getting \(\mathrm{x^2 + 6x - 5}\) instead of \(\mathrm{x^2 + 6x + 5}\)) or incorrect factoring. These errors lead to wrong solutions entirely.

This leads to confusion and guessing among the answer choices.

The Bottom Line:

This problem tests whether students can systematically solve a system involving both linear and quadratic equations, requiring them to find all solutions and then apply the constraint of selecting the greatest value.