To analyze the statements regarding the slopes and intercepts of the given lines, we will rewrite each line in slope-intercept form (y = mx + b) and identify their slopes and y-intercepts.
Line 1: 2x - 4y = 16
First, rearranging to get y alone: \[ -4y = -2x + 16 \ y = \frac{1}{2}x - 4 \]
- Slope (m) = 1/2
- Y-intercept (b) = -4
Line 2: y = -\frac{3}{4}x - 3
This line is already in slope-intercept form.
- Slope (m) = -3/4
- Y-intercept (b) = -3
Line 3: y - 5 = -2(x + 1)
Expanding this: \[ y - 5 = -2x - 2 \ y = -2x + 3 \]
- Slope (m) = -2
- Y-intercept (b) = 3
Comparison of the slopes:
- Line 1: Slope = 1/2
- Line 2: Slope = -3/4
- Line 3: Slope = -2
The slope value indicates steepness. A positive slope is less steep than a negative slope with a greater absolute value; thus:
- Line 1 has the least steep slope (1/2).
- Line 2 is steeper than Line 1 (since -3/4 is more negative).
- Line 3 is the steepest (the most negative slope of -2).
Comparison of the y-intercepts:
- Line 1: Y-intercept = -4
- Line 2: Y-intercept = -3
- Line 3: Y-intercept = 3
The greatest y-intercept is 3 from Line 3.
Now we can evaluate the statements based on our analysis:
- Line 3 has the least steep slope. (False, Line 1 has the least steep slope)
- Line 1 has the steepest slope. (False, Line 3 has the steepest slope)
- Line 1 has the biggest y-intercept. (False, Line 3 has the biggest y-intercept)
- Line 3 has a decreasing graph. (True, because slope is -2)
- Line 2 has an increasing graph. (False, slope is -3/4, so this line is decreasing.)
- Line 2 has the biggest y-intercept. (False, Line 3 has the biggest y-intercept)
Conclusion: The true statement is: Line 3 has a decreasing graph.