To determine the solution to the system of linear equations, let's first rewrite the equations clearly:
- \( 5t + 2y = -4 \)
- \( 5t + 2g = 1 \)
It appears there might be a typographical error with the variables. We'll assume the equations are correctly written and represent a system of equations in two variables.
We can analyze the two equations to see if they are consistent (have a solution) or if they are parallel (no solution).
Step 1: Write the equations clearly and simplify
- \( 5t + 2y = -4 \)
- \( 5t + 2g = 1 \)
Step 2: Express one variable in terms of the other.
From the first equation: \[ 5t = -4 - 2y \] \[ t = \frac{-4 - 2y}{5} \]
From the second equation: \[ 5t = 1 - 2g \] \[ t = \frac{1 - 2g}{5} \]
Step 3: Set the two equations for \( t \) equal to each other since both represent \( t \):
\[ \frac{-4 - 2y}{5} = \frac{1 - 2g}{5} \]
Step 4: Eliminate the denominator:
Multiply both sides by 5 (assuming \( 5 \neq 0 \)): \[ -4 - 2y = 1 - 2g \]
Step 5: Rearrange the equation:
\[ -2y + 2g = 1 + 4 \] \[ -2y + 2g = 5 \] \[ - y + g = \frac{5}{2} \]
Step 6: Check the consistency of the equations.
For the system to have a unique solution, the slopes and intercepts must differ. However, if we notice the form after rearranging, we find that \( g = y + \frac{5}{2} \), signifying a relationship rather than distinct solutions.
Step 7: Determine if there's no solution.
To evaluate further:
- If we try to set \( t \) to a value (say \( t = 0 \)), we get:
- From \( 5t + 2y = -4 \): \( 2y = -4 \) -> \( y = -2 \)
- From \( 5t + 2g = 1 \): \( 2g = 1 \) -> \( g = 0.5 \)
We found different y and g values for the same t. Thus, the lines are non-parallel.
Conclusion:
Since there are different values for \( y \) and \( g \) for the same value of \( t \), the system has infinitely many solutions.
None of the options provided clearly describe the solution set of the system. However, based on the analysis of the linear equations, the statement that stands correct regarding the situation is:
• A. The system has no solution. (Although they have inconsistent solutions, it's not parallel; thus, they can yield different values.)
Overall, the correct interpretation of the equations indicates that they express a consistent relationship leading to infinite values for combinations of solutions.
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