y = x^2 + 3x - 7y - 5x + 8 = 0How many solutions are there to the system...

1. TRANSLATE the problem information

• Given system:
  • \(\mathrm{y = x^2 + 3x - 7}\) (quadratic equation)
  • \(\mathrm{y - 5x + 8 = 0}\) (linear equation)
• Need to find: number of solutions to this system

2. INFER the solution approach

• Since both equations equal y, substitution is the most efficient method
• First, isolate y in the simpler (linear) equation
• Then substitute that expression for y into the quadratic equation

3. SIMPLIFY the linear equation

• From \(\mathrm{y - 5x + 8 = 0}\)
• Rearrange to get: \(\mathrm{y = 5x - 8}\)

4. INFER the substitution step

• Replace y in the first equation with \(\mathrm{(5x - 8)}\):
• \(\mathrm{5x - 8 = x^2 + 3x - 7}\)

5. SIMPLIFY to standard quadratic form

• Move all terms to one side:
  \(\mathrm{5x - 8 = x^2 + 3x - 7}\)
  \(\mathrm{0 = x^2 + 3x - 7 - 5x + 8}\)
  \(\mathrm{0 = x^2 - 2x + 1}\)

6. INFER the factoring approach

• Recognize this as a perfect square trinomial
• Factor: \(\mathrm{0 = (x - 1)^2}\)

7. SIMPLIFY to find x-value

• Since \(\mathrm{(x - 1)^2 = 0}\), we get \(\mathrm{x - 1 = 0}\)
• Therefore: \(\mathrm{x = 1}\)

8. SIMPLIFY to find corresponding y-value

• Substitute \(\mathrm{x = 1}\) into \(\mathrm{y = 5x - 8}\):
• \(\mathrm{y = 5(1) - 8 = -3}\)

9. INFER the final answer

• The system has exactly one solution: \(\mathrm{(1, -3)}\)
• This makes sense because a quadratic (parabola) and linear (line) equation can intersect at most at 2 points, and our quadratic gave us a repeated root

Answer: C. There is exactly 1 solution.

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY skill: Students make algebraic errors when combining like terms after substitution.

When rearranging \(\mathrm{5x - 8 = x^2 + 3x - 7}\), they might incorrectly combine:

• The x-terms: writing \(\mathrm{5x + 3x = 8x}\) instead of \(\mathrm{5x - 3x = 2x}\)
• The constants: writing \(\mathrm{-8 - 7 = -15}\) instead of \(\mathrm{-8 + 7 = -1}\)

This leads to an incorrect quadratic like \(\mathrm{x^2 + 8x - 15 = 0}\), which has two distinct solutions rather than one repeated solution.

This may lead them to select Choice B (There are exactly 2 solutions).

Second Most Common Error:

Poor INFER reasoning about solution interpretation: Students solve correctly to get \(\mathrm{(x - 1)^2 = 0}\) but don't recognize that this represents a repeated root, meaning only one solution point.

They see the equation \(\mathrm{(x - 1)^2 = 0}\) and think "since it's squared, there must be two solutions" without realizing that \(\mathrm{x = 1}\) is a double root corresponding to a single point of intersection.

This may lead them to select Choice B (There are exactly 2 solutions).

The Bottom Line:

This problem tests whether students can execute substitution carefully and understand that the number of solutions corresponds to intersection points, not just the degree of the resulting polynomial. The key insight is recognizing that a perfect square factor means the line is tangent to the parabola at exactly one point.