Question:Line m has y-intercept 7 and slope -3.Line n has slope 1/2 and x-intercept 4.What is the point of intersection...

1. TRANSLATE the problem information

• Given information:
  • Line m: y-intercept = 7, slope = -3
  • Line n: slope = 1/2, x-intercept = 4
  • Need to find: intersection point (x, y)

2. TRANSLATE each line description into an equation

For line m: Since we have slope and y-intercept directly:

• \(\mathrm{y = -3x + 7}\)

For line n: We have slope and x-intercept (passes through point (4, 0)):

• Using point-slope form: \(\mathrm{y - 0 = \frac{1}{2}(x - 4)}\)
• SIMPLIFY: \(\mathrm{y = \frac{1}{2}x - 2}\)

3. INFER the solution strategy

• At the intersection point, both equations must be satisfied
• This means: \(\mathrm{-3x + 7 = \frac{1}{2}x - 2}\)

4. SIMPLIFY to solve for x

• Start with: \(\mathrm{-3x + 7 = \frac{1}{2}x - 2}\)
• Add 3x to both sides: \(\mathrm{7 = \frac{1}{2}x + 3x - 2}\)
• Add 2 to both sides: \(\mathrm{9 = \frac{1}{2}x + 3x}\)
• Combine terms: \(\mathrm{9 = \frac{7}{2}x}\)
• Solve: \(\mathrm{x = 9 ÷ \frac{7}{2} = 9 × \frac{2}{7} = \frac{18}{7}}\)

5. SIMPLIFY to find y-coordinate

• Substitute \(\mathrm{x = \frac{18}{7}}\) into either equation (using line n's equation):
• \(\mathrm{y = \frac{1}{2}(\frac{18}{7}) - 2 = \frac{9}{7} - \frac{14}{7} = -\frac{5}{7}}\)

Answer: \(\mathrm{(\frac{18}{7}, -\frac{5}{7})}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Incorrectly writing the equation for line n from the x-intercept information.

Students might write \(\mathrm{y = \frac{1}{2}x + 4}\) instead of \(\mathrm{y = \frac{1}{2}x - 2}\), confusing the x-intercept with the y-intercept or not properly applying point-slope form. They know the line passes through \(\mathrm{(4, 0)}\) but incorrectly assume this means the y-intercept is 4.

This leads to solving \(\mathrm{-3x + 7 = \frac{1}{2}x + 4}\), giving \(\mathrm{x = \frac{6}{7}}\) and \(\mathrm{y = \frac{31}{7}}\), causing them to select Choice C \(\mathrm{(\frac{6}{7}, \frac{31}{7})}\).

Second Most Common Error:

Poor SIMPLIFY execution: Making arithmetic errors when combining fractions and negative numbers during algebraic manipulation.

Students correctly set up \(\mathrm{-3x + 7 = \frac{1}{2}x - 2}\) but make errors like incorrectly combining \(\mathrm{\frac{1}{2}x + 3x}\) or making sign errors when moving terms. These computational mistakes lead to wrong values for x and subsequently y.

This causes confusion about which answer choice matches their incorrect calculations, leading to guessing among the remaining options.

The Bottom Line:

Success requires careful translation of intercept information into proper equation form, followed by systematic algebraic manipulation with attention to fraction arithmetic.