Question

11
which equation is the best approximate for the line?
( y = \frac{1}{3}x - 1.7 )
( y = \frac{2}{3}x - 1.7 )
( y = x + 1.7 )
( y = x + 1.7 )

Explanation:

Step1: Identify slope and y-intercept

The line equation is \( y = mx + b \), where \( m \) is slope and \( b \) is y-intercept. From the graph, the line crosses y-axis at \( y \approx -1.7 \), so \( b \approx -1.7 \). Now calculate slope: take two points, e.g., (0, -1.7) and (3, -1). Slope \( m = \frac{-1 - (-1.7)}{3 - 0} = \frac{0.7}{3} \approx \frac{1}{3} \) (since \( \frac{1}{3} \approx 0.33 \), close to 0.7/3 ≈ 0.23? Wait, maybe better points. Wait, when \( x = 3 \), \( y \approx -1 \); \( x = 0 \), \( y \approx -1.7 \). So \( m = \frac{-1 - (-1.7)}{3 - 0} = \frac{0.7}{3} \approx 0.23 \), and \( \frac{1}{3} \approx 0.33 \), \( \frac{2}{3} \approx 0.66 \). Wait, maybe the line passes through (3, -1) and (0, -1.7). So slope \( m = \frac{-1 - (-1.7)}{3 - 0} = \frac{0.7}{3} \approx 0.23 \), which is closer to \( \frac{1}{3} \approx 0.33 \) than \( \frac{2}{3} \). Also, y-intercept is -1.7. So the equation \( y = \frac{1}{3}x - 1.7 \) matches.

Step2: Check options

• Option 1: \( y = \frac{1}{3}x - 1.7 \) – slope ~0.33, y-intercept -1.7. Matches.
• Option 2: \( y = \frac{2}{3}x - 1.7 \) – slope ~0.66, steeper than graph.
• Option 3: \( y = x + 1.7 \) – slope 1, positive y-intercept, wrong.
• Option 4: Same as 3, wrong.

Answer:

\( y = \frac{1}{3}x - 1.7 \) (the first option, with the red dot)