Question
Trang graphed the system of equations below.
y = negative 2 (x + 3). Y = negative 2 x + 3
Trang says that the system of equations has no solution. Which explains whether or not he is correct?
A. Trang is correct because the lines have the same slope but different y-intercepts.
B. Trang is correct because the lines have the same slope and the same y-intercept.
C. Trang is not correct because the lines have the same slope but different y-intercepts.
D. Trang is not correct because the lines have the same slope and the same y-intercept.
y = negative 2 (x + 3). Y = negative 2 x + 3
Trang says that the system of equations has no solution. Which explains whether or not he is correct?
A. Trang is correct because the lines have the same slope but different y-intercepts.
B. Trang is correct because the lines have the same slope and the same y-intercept.
C. Trang is not correct because the lines have the same slope but different y-intercepts.
D. Trang is not correct because the lines have the same slope and the same y-intercept.
Answers
GPT-4o mini
To determine whether Trang is correct about the system of equations having no solution, we can analyze the given equations:
1. **First equation:** \( y = -2(x + 3) \) can be rewritten:
\[
y = -2x - 6
\]
This has a slope of \(-2\) and a y-intercept of \(-6\).
2. **Second equation:** \( y = -2x + 3 \) has a slope of \(-2\) and a y-intercept of \(3\).
Now we can compare the two equations:
- Both lines have the **same slope** of \(-2\).
- However, they have **different y-intercepts** (\(-6\) for the first equation and \(3\) for the second).
Lines that have the same slope but different y-intercepts are **parallel** and will never intersect, which means they have **no solution**.
Hence, the correct explanation for Trang's statement is:
**A. Trang is correct because the lines have the same slope but different y-intercepts.**
1. **First equation:** \( y = -2(x + 3) \) can be rewritten:
\[
y = -2x - 6
\]
This has a slope of \(-2\) and a y-intercept of \(-6\).
2. **Second equation:** \( y = -2x + 3 \) has a slope of \(-2\) and a y-intercept of \(3\).
Now we can compare the two equations:
- Both lines have the **same slope** of \(-2\).
- However, they have **different y-intercepts** (\(-6\) for the first equation and \(3\) for the second).
Lines that have the same slope but different y-intercepts are **parallel** and will never intersect, which means they have **no solution**.
Hence, the correct explanation for Trang's statement is:
**A. Trang is correct because the lines have the same slope but different y-intercepts.**