Questions: Solve the system by elimination. If the system is inconsistent or has dependent equations, say so. 3x+7y=2 -6x-14y=-4 Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The system has a single solution. The solution set is . (Type an ordered pair. Type integers or simplified fractions.) B. There are infinitely many solutions and the equations are dependent. The solution set is (x, y) 3x+7x=2. C. The system is inconsistent. The solution set is the empty set.

Solution Steps

To solve the system of equations by elimination, we first observe that the second equation is a multiple of the first equation. This indicates that the two equations are dependent, meaning they represent the same line. Therefore, the system has infinitely many solutions.

We are given the following system of equations:

\[ \begin{array}{c} 3x + 7y = 2 \\ -6x - 14y = -4 \end{array} \]

To eliminate one of the variables, we can multiply the first equation by 2 to make the coefficients of \(x\) in both equations equal in magnitude:

\[ 2(3x + 7y) = 2 \times 2 \]

This gives us:

\[ 6x + 14y = 4 \]

Now, we add the modified first equation to the second equation:

\[ \begin{array}{c} 6x + 14y = 4 \\ -6x - 14y = -4 \end{array} \]

Adding these equations results in:

\[ (6x - 6x) + (14y - 14y) = 4 - 4 \]

\[ 0 = 0 \]

The result \(0 = 0\) is a true statement, indicating that the two equations are dependent. This means that they represent the same line, and thus, there are infinitely many solutions.

Final Answer