Question
Use the interactive graph to produce an orange line identical to the blue line. Then change the value of b only.
Which equation would be parallel to, and have no points in common with, y = 2(x + 3)?
y = x + 6
y = –2x –6
y = 2x – 3
y = 2x + 6
Answers
Answered by
GPT-4o mini
To determine which equation is parallel to \( y = 2(x + 3) \), we first need to recognize the characteristics of parallel lines. Two lines are parallel if they have the same slope but different y-intercepts.
The given equation \( y = 2(x + 3) \) can be rewritten in slope-intercept form:
\[
y = 2x + 6
\]
From this, we see that the slope is 2. Now, we need to evaluate the options provided to see which one has the same slope of 2:
1. \( y = x + 6 \) → slope is 1
2. \( y = -2x - 6 \) → slope is -2
3. \( y = 2x - 3 \) → slope is 2 (but shares a point)
4. \( y = 2x + 6 \) → slope is 2 (and is the same line)
Thus, the only equation that is parallel (same slope) but has a different y-intercept is:
- **None of the given equations satisfy the criteria.**
Since the option \( y = 2x + 6 \) is the same line, and \( y = 2x - 3 \) intersects, it's clear that there are no parallel options given as per the question.
For a parallel equation, it would need to be something like \( y = 2x + c \) where \( c \) is not 6.
If we were to choose among the provided options and needing an answer based solely on similar slope but not listed:
- **No correct answer provided.**
To find a valid option, you'd need an equation like \( y = 2x + c \) where \( c \) does not equal 6 for it to be truly parallel and not intersecting with the original line.
The given equation \( y = 2(x + 3) \) can be rewritten in slope-intercept form:
\[
y = 2x + 6
\]
From this, we see that the slope is 2. Now, we need to evaluate the options provided to see which one has the same slope of 2:
1. \( y = x + 6 \) → slope is 1
2. \( y = -2x - 6 \) → slope is -2
3. \( y = 2x - 3 \) → slope is 2 (but shares a point)
4. \( y = 2x + 6 \) → slope is 2 (and is the same line)
Thus, the only equation that is parallel (same slope) but has a different y-intercept is:
- **None of the given equations satisfy the criteria.**
Since the option \( y = 2x + 6 \) is the same line, and \( y = 2x - 3 \) intersects, it's clear that there are no parallel options given as per the question.
For a parallel equation, it would need to be something like \( y = 2x + c \) where \( c \) is not 6.
If we were to choose among the provided options and needing an answer based solely on similar slope but not listed:
- **No correct answer provided.**
To find a valid option, you'd need an equation like \( y = 2x + c \) where \( c \) does not equal 6 for it to be truly parallel and not intersecting with the original line.