Question
y=23x-1
y=23x-2
one solution, no solution, or infinite solutions
y=23x-2
one solution, no solution, or infinite solutions
Answers
GPT-4o mini
To determine the number of solutions for the system of equations given by \( y = 23x - 1 \) and \( y = 23x - 2 \), we can analyze the equations.
1. The first equation is \( y = 23x - 1 \).
2. The second equation is \( y = 23x - 2 \).
Both equations are linear and have the same slope (23). We can rewrite them:
- The first equation can be rearranged to the standard form:
\[
23x - y - 1 = 0
\]
- The second equation can be rearranged similarly:
\[
23x - y - 2 = 0
\]
Since both lines have the same slope (indicating they are parallel), we need to ascertain whether they are distinct lines or the same line. To do this, we can compare the y-intercepts:
- The y-intercept of the first line is -1.
- The y-intercept of the second line is -2.
Because the y-intercepts are different, the two lines are parallel but will never intersect.
### Conclusion:
The system of equations has **no solution** because the lines are parallel and distinct.
1. The first equation is \( y = 23x - 1 \).
2. The second equation is \( y = 23x - 2 \).
Both equations are linear and have the same slope (23). We can rewrite them:
- The first equation can be rearranged to the standard form:
\[
23x - y - 1 = 0
\]
- The second equation can be rearranged similarly:
\[
23x - y - 2 = 0
\]
Since both lines have the same slope (indicating they are parallel), we need to ascertain whether they are distinct lines or the same line. To do this, we can compare the y-intercepts:
- The y-intercept of the first line is -1.
- The y-intercept of the second line is -2.
Because the y-intercepts are different, the two lines are parallel but will never intersect.
### Conclusion:
The system of equations has **no solution** because the lines are parallel and distinct.