Question
Roxanne graphed the system of equations to find the solution.
y = 2/3x - 5 and y = -2x + 3
She determined that the solution is (-3, -3). Is she correct? If not, choose the correct explanation
A. No, she used the wrong slopes when graphing the equations.
B. Yes, she is correct.
C. No, she used the wrong y-intercepts when graphing the equations.
D. No, she switched the x and y values of the intersection point when writing the solution.
y = 2/3x - 5 and y = -2x + 3
She determined that the solution is (-3, -3). Is she correct? If not, choose the correct explanation
A. No, she used the wrong slopes when graphing the equations.
B. Yes, she is correct.
C. No, she used the wrong y-intercepts when graphing the equations.
D. No, she switched the x and y values of the intersection point when writing the solution.
Answers
GPT-4o mini
To determine if Roxanne's solution of (-3, -3) is correct, we need to substitute the x-value and y-value into both equations to see if they satisfy both.
1. For the first equation \(y = \frac{2}{3}x - 5\):
Substitute \(x = -3\):
\[
y = \frac{2}{3}(-3) - 5 = -2 - 5 = -7
\]
This means that when \(x = -3\), \(y\) should be \(-7\), not \(-3\).
2. For the second equation \(y = -2x + 3\):
Substitute \(x = -3\):
\[
y = -2(-3) + 3 = 6 + 3 = 9
\]
This means that when \(x = -3\), \(y\) should be \(9\), not \(-3\).
Since the point (-3, -3) does not satisfy either equation, Roxanne is incorrect.
Now let's evaluate the provided choices:
- A. No, she used the wrong slopes when graphing the equations. (This may not be the case; we haven't checked the slopes, just the intersection.)
- B. Yes, she is correct. (This is not true.)
- C. No, she used the wrong y-intercepts when graphing the equations. (We haven't checked this specifically.)
- D. No, she switched the x and y values of the intersection point when writing the solution. (This is possible since her answer seems off based on substitution.)
Given the evaluations above, the most plausible explanation would be:
**D. No, she switched the x and y values of the intersection point when writing the solution.**
This implies that she might have written the coordinates in the incorrect order resulting in an incorrect solution.
1. For the first equation \(y = \frac{2}{3}x - 5\):
Substitute \(x = -3\):
\[
y = \frac{2}{3}(-3) - 5 = -2 - 5 = -7
\]
This means that when \(x = -3\), \(y\) should be \(-7\), not \(-3\).
2. For the second equation \(y = -2x + 3\):
Substitute \(x = -3\):
\[
y = -2(-3) + 3 = 6 + 3 = 9
\]
This means that when \(x = -3\), \(y\) should be \(9\), not \(-3\).
Since the point (-3, -3) does not satisfy either equation, Roxanne is incorrect.
Now let's evaluate the provided choices:
- A. No, she used the wrong slopes when graphing the equations. (This may not be the case; we haven't checked the slopes, just the intersection.)
- B. Yes, she is correct. (This is not true.)
- C. No, she used the wrong y-intercepts when graphing the equations. (We haven't checked this specifically.)
- D. No, she switched the x and y values of the intersection point when writing the solution. (This is possible since her answer seems off based on substitution.)
Given the evaluations above, the most plausible explanation would be:
**D. No, she switched the x and y values of the intersection point when writing the solution.**
This implies that she might have written the coordinates in the incorrect order resulting in an incorrect solution.