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Sagot :
Let's analyze and solve the problem step-by-step to determine which statement about the graphs is true.
### Step 1: Identify the slope and intercept of the original function
The original function is given by:
[tex]\[ y = \frac{B}{5} x + 4 \][/tex]
To find the slope, we need to look at the coefficient of [tex]\( x \)[/tex]:
[tex]\[ \text{slope of the original function} = \frac{B}{5} \][/tex]
The y-intercept of the original function is:
[tex]\[ 4 \][/tex]
### Step 2: Identify the slope and intercept of the new function
The new function is given:
[tex]\[ y = -\frac{5}{8} x + 8 \][/tex]
The slope is:
[tex]\[ -\frac{5}{8} \][/tex]
The y-intercept of the new function is:
[tex]\[ 8 \][/tex]
### Step 3: Determine if the lines are parallel
Two lines are parallel if their slopes are equal. So we compare the slopes:
[tex]\[ \frac{B}{5} \text{ and } -\frac{5}{8} \][/tex]
Clearly, the two slopes are not equal unless:
[tex]\[ \frac{B}{5} = -\frac{5}{8} \][/tex]
This equation has no integer solution for [tex]\( B \)[/tex], so the lines are not parallel.
### Step 4: Determine if the lines are perpendicular
Two lines are perpendicular if the product of their slopes is [tex]\(-1\)[/tex]:
[tex]\[ \left(\frac{B}{5}\right) \times \left(-\frac{5}{8}\right) = -1 \][/tex]
Simplifying the product:
[tex]\[ \frac{-5B}{40} = -1 \][/tex]
[tex]\[ \frac{-B}{8} = -1 \][/tex]
[tex]\[ B = 8 \][/tex]
So, if [tex]\( B = 8 \)[/tex], then the two lines are perpendicular.
### Step 5: Identify intersection points
The lines intersect where:
[tex]\[ \frac{B}{5}x + 4 = -\frac{5}{8}x + 8 \][/tex]
Substituting [tex]\( B = 8 \)[/tex]:
[tex]\[ \frac{8}{5}x + 4 = -\frac{5}{8}x + 8 \][/tex]
To find [tex]\( x \)[/tex]:
[tex]\[ \frac{8}{5}x + \frac{5}{8}x = 8 - 4 \][/tex]
[tex]\[ (\frac{64}{40} + \frac{25}{40}) x = 4 \][/tex]
[tex]\[ \frac{89}{40} x = 4 \][/tex]
[tex]\[ x = \frac{4 \times 40}{89} = \frac{160}{89} \][/tex]
Substituting [tex]\( x \)[/tex] back to find [tex]\( y \)[/tex]:
[tex]\[ y = \frac{8}{5} \times \frac{160}{89} + 4 \][/tex]
[tex]\[ y = \left( \frac{1280}{445} + \frac{1780}{445} \right) \][/tex]
[tex]\[ y = 4 \][/tex]
Since we have calculated [tex]\( x \approx 1.8 \)[/tex], clearly it doesn’t satisfy our intersection points at specific options a and intersection at 0,8 also doesn’t hold true for [tex]\( x = 0 \)[/tex].
### Conclusion:
After analyzing all options and accounting for the perpendicular nature derived when [tex]\( B=8 \)[/tex], we have:
B. The graph of the original function is perpendicular to the graph of the new function is correct answer.
### Step 1: Identify the slope and intercept of the original function
The original function is given by:
[tex]\[ y = \frac{B}{5} x + 4 \][/tex]
To find the slope, we need to look at the coefficient of [tex]\( x \)[/tex]:
[tex]\[ \text{slope of the original function} = \frac{B}{5} \][/tex]
The y-intercept of the original function is:
[tex]\[ 4 \][/tex]
### Step 2: Identify the slope and intercept of the new function
The new function is given:
[tex]\[ y = -\frac{5}{8} x + 8 \][/tex]
The slope is:
[tex]\[ -\frac{5}{8} \][/tex]
The y-intercept of the new function is:
[tex]\[ 8 \][/tex]
### Step 3: Determine if the lines are parallel
Two lines are parallel if their slopes are equal. So we compare the slopes:
[tex]\[ \frac{B}{5} \text{ and } -\frac{5}{8} \][/tex]
Clearly, the two slopes are not equal unless:
[tex]\[ \frac{B}{5} = -\frac{5}{8} \][/tex]
This equation has no integer solution for [tex]\( B \)[/tex], so the lines are not parallel.
### Step 4: Determine if the lines are perpendicular
Two lines are perpendicular if the product of their slopes is [tex]\(-1\)[/tex]:
[tex]\[ \left(\frac{B}{5}\right) \times \left(-\frac{5}{8}\right) = -1 \][/tex]
Simplifying the product:
[tex]\[ \frac{-5B}{40} = -1 \][/tex]
[tex]\[ \frac{-B}{8} = -1 \][/tex]
[tex]\[ B = 8 \][/tex]
So, if [tex]\( B = 8 \)[/tex], then the two lines are perpendicular.
### Step 5: Identify intersection points
The lines intersect where:
[tex]\[ \frac{B}{5}x + 4 = -\frac{5}{8}x + 8 \][/tex]
Substituting [tex]\( B = 8 \)[/tex]:
[tex]\[ \frac{8}{5}x + 4 = -\frac{5}{8}x + 8 \][/tex]
To find [tex]\( x \)[/tex]:
[tex]\[ \frac{8}{5}x + \frac{5}{8}x = 8 - 4 \][/tex]
[tex]\[ (\frac{64}{40} + \frac{25}{40}) x = 4 \][/tex]
[tex]\[ \frac{89}{40} x = 4 \][/tex]
[tex]\[ x = \frac{4 \times 40}{89} = \frac{160}{89} \][/tex]
Substituting [tex]\( x \)[/tex] back to find [tex]\( y \)[/tex]:
[tex]\[ y = \frac{8}{5} \times \frac{160}{89} + 4 \][/tex]
[tex]\[ y = \left( \frac{1280}{445} + \frac{1780}{445} \right) \][/tex]
[tex]\[ y = 4 \][/tex]
Since we have calculated [tex]\( x \approx 1.8 \)[/tex], clearly it doesn’t satisfy our intersection points at specific options a and intersection at 0,8 also doesn’t hold true for [tex]\( x = 0 \)[/tex].
### Conclusion:
After analyzing all options and accounting for the perpendicular nature derived when [tex]\( B=8 \)[/tex], we have:
B. The graph of the original function is perpendicular to the graph of the new function is correct answer.
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