Question
Suppose the first equation in a system of two linear equations is 12x + 7y = 25. The second equation being which of these will cause the system to have no solution?
A.
12x + 7y = 20
B.
12x + 7y = 25
C.
12x + 9y = 20
D.
12x + 9y = 25
A.
12x + 7y = 20
B.
12x + 7y = 25
C.
12x + 9y = 20
D.
12x + 9y = 25
Answers
GPT-4o mini
In order for a system of two linear equations to have no solution, the two equations must be parallel. This occurs when the coefficients of \( x \) and \( y \) are proportional, but the constants on the right-hand side are not.
The first equation given is:
\[ 12x + 7y = 25. \]
Now, let's analyze each option:
**A.** \( 12x + 7y = 20 \)
This equation has the same coefficients as the first equation. The lines are parallel, but since they have different constants (25 ≠ 20), this will cause the system to have no solution.
**B.** \( 12x + 7y = 25 \)
This equation is identical to the first equation. So, they represent the same line, and thus, the system has infinitely many solutions.
**C.** \( 12x + 9y = 20 \)
The coefficients of \( y \) (7 and 9) are not proportional to the coefficients of \( x \) (12), so these lines are not parallel. This option will result in one point of intersection and thus has a unique solution.
**D.** \( 12x + 9y = 25 \)
Similar to option C, the coefficients of \( y \) (7 and 9) are not proportional, so these lines are not parallel. This option will also result in a unique solution.
Based on this analysis, the only equation that will cause the system to have no solution is:
**A. \( 12x + 7y = 20 \)**.
The first equation given is:
\[ 12x + 7y = 25. \]
Now, let's analyze each option:
**A.** \( 12x + 7y = 20 \)
This equation has the same coefficients as the first equation. The lines are parallel, but since they have different constants (25 ≠ 20), this will cause the system to have no solution.
**B.** \( 12x + 7y = 25 \)
This equation is identical to the first equation. So, they represent the same line, and thus, the system has infinitely many solutions.
**C.** \( 12x + 9y = 20 \)
The coefficients of \( y \) (7 and 9) are not proportional to the coefficients of \( x \) (12), so these lines are not parallel. This option will result in one point of intersection and thus has a unique solution.
**D.** \( 12x + 9y = 25 \)
Similar to option C, the coefficients of \( y \) (7 and 9) are not proportional, so these lines are not parallel. This option will also result in a unique solution.
Based on this analysis, the only equation that will cause the system to have no solution is:
**A. \( 12x + 7y = 20 \)**.