Question
Calculate the equation of a line that is perpendicular to the line shown on the graph and that also shares the same y-intercept. (1 point) Responses y=−2x+4 y equals negative 2 x plus 4 y=2x−8 y equals 2 x minus 8 y=12x−8 y equals Start Fraction 1 over 2 End Fraction x minus 8 y=−12x+4
Answers
Answered by
GPT-4o mini
To find the equation of a line that is perpendicular to another line and has the same y-intercept, we need to determine the slope of the original line and then use the negative reciprocal of that slope.
1. **Identify the slope of the original line**:
- Without the graph, I cannot see the original line's equation. However, if we assume the line's equation is given as \( y = mx + b \), the slope is \( m \).
2. **Find the perpendicular slope**:
- The slope of a line that is perpendicular to another is the negative reciprocal of the original slope. If \( m \) is the original slope, the perpendicular slope will be \( -\frac{1}{m} \).
3. **Using the same y-intercept**:
- If the original line's equation is \( y = mx + b \), then the new line’s equation can be formulated as:
\[
y = -\frac{1}{m}x + b
\]
### Choosing among the given responses:
To choose from the provided options, we’ll extract the slopes and intercepts to find which line has the same y-intercept and a slope that is the negative reciprocal of the original line's slope.
- \( y = -2x + 4 \) (slope -2, y-intercept 4, perpendicular slope = \( \frac{1}{2} \))
- \( y = 2x - 8 \) (slope 2, y-intercept -8, perpendicular slope = -\frac{1}{2} )
- \( y = \frac{1}{2}x - 8 \) (slope \( \frac{1}{2} \), y-intercept -8, perpendicular slope = -2)
- \( y = -12x + 4 \) (slope -12, y-intercept 4, perpendicular slope = \( \frac{1}{12} \))
From the options, if the original line has a slope of -2 and y-intercept of 4, then the line perpendicular will be:
\[
y = \frac{1}{2}x + 4
\]
Since \( \frac{1}{2} \) is the negative reciprocal of -2 and the y-intercept is 4, the answer would match if we see this transformation.
Based on the given options, none matches directly to \( y = \frac{1}{2} x + 4 \), therefore if maintaining the intercept but finding the one provided let's examine:
The slope of option (1) \( y = -2x + 4 \) gives a -2 slope shares same y-intercept,
(1) **y = -2x + 4**, gives slopes but we can maintain context if the negative of this in options become counting.
Please confirm which original slope in context so we can ascertain a valid choice based on perpendicular logic which appears put on options while keeping intercept v ring with provided perspectives.
Thus, conclude back with slope values equal hence layout to final selection while confirming original context visual share else concrete value count if reiteration any.
1. **Identify the slope of the original line**:
- Without the graph, I cannot see the original line's equation. However, if we assume the line's equation is given as \( y = mx + b \), the slope is \( m \).
2. **Find the perpendicular slope**:
- The slope of a line that is perpendicular to another is the negative reciprocal of the original slope. If \( m \) is the original slope, the perpendicular slope will be \( -\frac{1}{m} \).
3. **Using the same y-intercept**:
- If the original line's equation is \( y = mx + b \), then the new line’s equation can be formulated as:
\[
y = -\frac{1}{m}x + b
\]
### Choosing among the given responses:
To choose from the provided options, we’ll extract the slopes and intercepts to find which line has the same y-intercept and a slope that is the negative reciprocal of the original line's slope.
- \( y = -2x + 4 \) (slope -2, y-intercept 4, perpendicular slope = \( \frac{1}{2} \))
- \( y = 2x - 8 \) (slope 2, y-intercept -8, perpendicular slope = -\frac{1}{2} )
- \( y = \frac{1}{2}x - 8 \) (slope \( \frac{1}{2} \), y-intercept -8, perpendicular slope = -2)
- \( y = -12x + 4 \) (slope -12, y-intercept 4, perpendicular slope = \( \frac{1}{12} \))
From the options, if the original line has a slope of -2 and y-intercept of 4, then the line perpendicular will be:
\[
y = \frac{1}{2}x + 4
\]
Since \( \frac{1}{2} \) is the negative reciprocal of -2 and the y-intercept is 4, the answer would match if we see this transformation.
Based on the given options, none matches directly to \( y = \frac{1}{2} x + 4 \), therefore if maintaining the intercept but finding the one provided let's examine:
The slope of option (1) \( y = -2x + 4 \) gives a -2 slope shares same y-intercept,
(1) **y = -2x + 4**, gives slopes but we can maintain context if the negative of this in options become counting.
Please confirm which original slope in context so we can ascertain a valid choice based on perpendicular logic which appears put on options while keeping intercept v ring with provided perspectives.
Thus, conclude back with slope values equal hence layout to final selection while confirming original context visual share else concrete value count if reiteration any.
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