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.

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

Transcript text: Solve the system by elimination. If the system is inconsistent or has dependent equations, say so. \[ \begin{array}{c} 3 x+7 y=2 \\ -6 x-14 y=-4 \end{array} \] 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 $\{(x, y) \mid 3 x+7 x=2\}$. C. The system is inconsistent. The solution set is the empty set.

Solution

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.

Step 1: Write Down the System of Equations

We are given the following system of equations:

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

Step 2: Multiply the First Equation

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:

\[ 6x + 14y = 4 \]

Step 3: Add the Equations

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:

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

\[ 0 = 0 \]

Step 4: Analyze the Result

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.