Question
The equation of Line 1: 2x−4y=16
The equation of Line 2: y=−34x−3
The equation of Line 3: y−5=−2(x+1)
Which statement is true?
(2 points)
Responses
Line 3 has a decreasing graph.
Line 3 has a decreasing graph.
Line 2 has an increasing graph.
Line 2 has an increasing graph.
Line 1 has the biggest y-intercept.
Line 1 has the biggest y-intercept.
Line 1 has the steepest slope.
Line 1 has the steepest slope.
Line 2 has the biggest y-intercept.
Line 2 has the biggest y-intercept.
Line 3 has the least steep slope.
Line 3 has the least steep slope.
Skip to navigation
The equation of Line 2: y=−34x−3
The equation of Line 3: y−5=−2(x+1)
Which statement is true?
(2 points)
Responses
Line 3 has a decreasing graph.
Line 3 has a decreasing graph.
Line 2 has an increasing graph.
Line 2 has an increasing graph.
Line 1 has the biggest y-intercept.
Line 1 has the biggest y-intercept.
Line 1 has the steepest slope.
Line 1 has the steepest slope.
Line 2 has the biggest y-intercept.
Line 2 has the biggest y-intercept.
Line 3 has the least steep slope.
Line 3 has the least steep slope.
Skip to navigation
Answers
GPT-4o mini
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.
1. **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
2. **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
3. **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.
- **Slopes:**
- Line 1: \(\frac{1}{2}\) (increasing)
- Line 2: -\(\frac{3}{4}\) (decreasing)
- Line 3: -2 (decreasing)
- **Y-intercepts:**
- Line 1: -4
- Line 2: -3
- Line 3: 3
Now let’s evaluate each statement:
1. **Line 3 has a decreasing graph:** True (slope is -2)
2. **Line 2 has an increasing graph:** False (slope is -3/4)
3. **Line 1 has the biggest y-intercept:** False (Line 3 has y-intercept 3, which is greater than Line 1's -4)
4. **Line 1 has the steepest slope:** False (Line 3's slope of -2 is steeper than Line 1's \(\frac{1}{2}\))
5. **Line 2 has the biggest y-intercept:** False (Line 3 has the biggest y-intercept)
6. **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.**
1. **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
2. **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
3. **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.
- **Slopes:**
- Line 1: \(\frac{1}{2}\) (increasing)
- Line 2: -\(\frac{3}{4}\) (decreasing)
- Line 3: -2 (decreasing)
- **Y-intercepts:**
- Line 1: -4
- Line 2: -3
- Line 3: 3
Now let’s evaluate each statement:
1. **Line 3 has a decreasing graph:** True (slope is -2)
2. **Line 2 has an increasing graph:** False (slope is -3/4)
3. **Line 1 has the biggest y-intercept:** False (Line 3 has y-intercept 3, which is greater than Line 1's -4)
4. **Line 1 has the steepest slope:** False (Line 3's slope of -2 is steeper than Line 1's \(\frac{1}{2}\))
5. **Line 2 has the biggest y-intercept:** False (Line 3 has the biggest y-intercept)
6. **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.**