Question

problem 5: select all expressions that could represent the slope of this line. \\(\frac{1}{4}\\) \\(\frac{4}{6}\\) \\(\frac{2}{3}\\) \\(\frac{8}{12}\\)

Explanation:

Step1: Recall Slope Formula

Slope \( m = \frac{\text{rise}}{\text{run}}=\frac{y_2 - y_1}{x_2 - x_1} \), and equivalent fractions (simplified or scaled) represent the same slope.

Step2: Simplify/Scale Each Option

• For \( \frac{1}{4} \): No obvious scaling to match others yet.
• For \( \frac{4}{6} \): Simplify to \( \frac{2}{3} \) (divide numerator/denominator by 2).
• For \( \frac{2}{3} \): Matches simplified \( \frac{4}{6} \).
• For \( \frac{8}{12} \): Simplify by dividing numerator/denominator by 4: \( \frac{2}{3} \) (wait, 8÷4=2, 12÷4=3? Wait, 8 and 12: GCD is 4? Wait 8÷4=2, 12÷4=3? No, 8 and 12 GCD is 4? Wait 8÷4=2, 12÷4=3? Wait no, 8 and 12: 8=2×4, 12=3×4, so \( \frac{8}{12}=\frac{2}{3} \) (divide by 4? Wait 8÷4=2, 12÷4=3? Yes. Wait also, if we scale \( \frac{2}{3} \) by 4: \( \frac{8}{12} \), or scale \( \frac{4}{6} \) by 2: \( \frac{8}{12} \)? Wait maybe I misread, but let's check each:

Wait the options (assuming the last one is \( \frac{8}{12} \) maybe? Wait the user's image: let's re-express. Wait the options are \( \frac{1}{4} \), \( \frac{4}{6} \), (blurred), \( \frac{2}{3} \), \( \frac{8}{12} \) (maybe). Wait \( \frac{4}{6} = \frac{2}{3} \) (divide by 2), \( \frac{8}{12} = \frac{2}{3} \) (divide by 4). So \( \frac{4}{6} \), \( \frac{2}{3} \), \( \frac{8}{12} \) are equivalent (since \( \frac{4}{6}=\frac{2}{3} \), \( \frac{8}{12}=\frac{2}{3} \)). Wait maybe the blurred one is not relevant, but the key is that \( \frac{4}{6} \), \( \frac{2}{3} \), \( \frac{8}{12} \) (if that's the last) are equivalent. Wait maybe the original problem's line has slope \( \frac{2}{3} \), so equivalent fractions: \( \frac{4}{6} \) (multiply numerator/denominator by 2), \( \frac{8}{12} \) (multiply by 4), and \( \frac{2}{3} \) itself. So the correct ones are \( \frac{4}{6} \), \( \frac{2}{3} \), \( \frac{8}{12} \) (assuming the last is \( \frac{8}{12} \)). Wait but let's confirm:

• \( \frac{4}{6} \): Simplify to \( \frac{2}{3} \) (divide numerator and denominator by 2).
• \( \frac{2}{3} \): Original simplified.
• \( \frac{8}{12} \): Simplify by dividing numerator and denominator by 4: \( \frac{2}{3} \), or multiply \( \frac{2}{3} \) by 4: \( \frac{8}{12} \).

So these three (\( \frac{4}{6} \), \( \frac{2}{3} \), \( \frac{8}{12} \)) are equivalent, representing the same slope. The \( \frac{1}{4} \) is different.

Answer:

The expressions representing the slope (assuming equivalent fractions) are:

• \( \frac{4}{6} \)
• \( \frac{2}{3} \)
• \( \frac{8}{12} \) (or the last option, depending on the image; but based on simplification, \( \frac{4}{6} = \frac{2}{3} \), \( \frac{8}{12} = \frac{2}{3} \), so these are equivalent to \( \frac{2}{3} \)).

(Note: If the last option is \( \frac{8}{12} \), then \( \frac{4}{6} \), \( \frac{2}{3} \), \( \frac{8}{12} \) are correct. If the blurred option is irrelevant, focus on the clear ones: \( \frac{4}{6} \), \( \frac{2}{3} \), and \( \frac{8}{12} \) (if that's the last) as they simplify to \( \frac{2}{3} \).)