We are given the following system of equations:

$x + 5y = 3$ $2x - 6 = -10y$

Step 1: Rewrite the equations in standard form.

First, rewrite both equations in the form $Ax + By = C$.

1. The first equation is already in standard form: $x + 5y = 3$

2. For the second equation, move everything involving variables to one side: $2x + 10y = -6$ (Note: We added $10y$ to both sides to combine like terms.)

So, the system becomes: $x + 5y = 3$ $2x + 10y = -6$

Step 2: Solve using determinants.

For a system of two equations: $Ax + By = C$ $Dx + Ey = F$ The determinant method uses the following formulas for $x$ and $y$:

$x = \frac{\text{determinant of} \begin{pmatrix} C & B \\ F & E \end{pmatrix}}{\text{determinant of} \begin{pmatrix} A & B \\ D & E \end{pmatrix}}$ $y = \frac{\text{determinant of} \begin{pmatrix} A & C \\ D & F \end{pmatrix}}{\text{determinant of} \begin{pmatrix} A & B \\ D & E \end{pmatrix}}$

The determinant of a 2x2 matrix: $\begin{pmatrix} a & b \\ c & d \end{pmatrix} = ad - bc$

Now, set up the matrices:

For the system: $\begin{pmatrix} 1 & 5 \\ 2 & 10 \end{pmatrix}$ The determinant of this matrix is: $\text{det} = 1(10) - 5(2) = 10 - 10 = 0$

Since the determinant is zero, the system is either inconsistent (no solution) or dependent (infinitely many solutions).

Step 3: Analyze the system.

Since the determinant is zero, the equations represent parallel lines that either don't intersect or are the same line.

Check the consistency by analyzing the relationship between the two equations:

• The second equation is simply a multiple of the first equation. Multiply the first equation by 2: $2(x + 5y) = 2(3)$ This gives: $2x + 10y = 6$ But we have $2x + 10y = -6$ from the second equation. This is a contradiction.

Thus, there is no solution to the system.

Final Answer:

The correct choice is: C. There is no solution.

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Would you like further clarification or details? Here are some related questions for further practice:

1. How can we determine if a system has infinitely many solutions using determinants?
2. What happens when the determinant of a system is non-zero?
3. Can two linear equations always represent two distinct lines?
4. How can you solve a system of equations using substitution or elimination methods?
5. What is the geometric interpretation of a system with no solution?

Tip: If the determinant of the coefficient matrix is zero, check if the equations are multiples of each other to see if they are dependent or inconsistent.