Bot GPT 4-o mini To determine if the system of equations has no solutions, infinitely many solutions, or exactly one solution, we can analyze the equations.
The given equations are:
- \( x + 5y = 2 \)
- \( 3x + 15y = 8 \)
We can manipulate the first equation to see if it relates to the second one. Let's multiply the first equation by 3 to see if the second equation is a multiple of the first:
\[ 3(x + 5y) = 3(2) \] This simplifies to:
\[ 3x + 15y = 6 \]
Now we can compare the results:
- From the first equation scaled by 3: \( 3x + 15y = 6 \)
- The second equation: \( 3x + 15y = 8 \)
Now we see that these two equations represent the same left-hand side, but different right-hand sides:
\[ 3x + 15y = 6 \quad \text{(from the first equation)} \] \[ 3x + 15y = 8 \quad \text{(the second equation)} \]
This means that the two equations represent parallel lines in the coordinate plane. Since the lines are parallel, they will never intersect.
Thus, the system of equations has no solutions.
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