To determine which statements are true regarding the slopes and y-intercepts of the lines, let's first rewrite each line in slope-intercept form (y = mx + b), where m is the slope and b is the y-intercept.
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Line 1: 2x - 4y = 16
Rearranging to slope-intercept form:
\[ -4y = -2x + 16 \] \[ y = \frac{1}{2}x - 4 \]
- Slope (m) = \( \frac{1}{2} \)
- Y-intercept (b) = -4
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Line 2: y = -\frac{3}{4}x - 3
This line is already in slope-intercept form.
- Slope (m) = -\(\frac{3}{4}\)
- Y-intercept (b) = -3
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Line 3: y - 5 = -2(x + 1)
Rearranging to slope-intercept form:
\[ y - 5 = -2x - 2 \] \[ y = -2x + 3 \]
- Slope (m) = -2
- Y-intercept (b) = 3
Now, let's analyze the information we've gathered.
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Slopes:
- Line 1: \(\frac{1}{2}\) (increasing)
- Line 2: -\(\frac{3}{4}\) (decreasing)
- Line 3: -2 (decreasing)
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Y-intercepts:
- Line 1: -4
- Line 2: -3
- Line 3: 3
Now let’s evaluate each statement:
- Line 3 has a decreasing graph: True (slope is -2)
- Line 2 has an increasing graph: False (slope is -3/4)
- Line 1 has the biggest y-intercept: False (Line 3 has y-intercept 3, which is greater than Line 1's -4)
- Line 1 has the steepest slope: False (Line 3's slope of -2 is steeper than Line 1's \(\frac{1}{2}\))
- Line 2 has the biggest y-intercept: False (Line 3 has the biggest y-intercept)
- Line 3 has the least steep slope: True (Line 1 has the least steep slope of \(\frac{1}{2}\), but since Line 3's slope is -2 and Line 1's slope is positive, the correct sense for least steep means comparing their absolute values; Line 3 is the least steep among the negatively sloped lines)
Given this analysis, the true statements are:
- Line 3 has a decreasing graph.
- Line 3 has the least steep slope.
So, the most correct single true statement is:
Line 3 has a decreasing graph.
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