Solve the system of equations using matrix method:2x + 3y = 5;4x + 6y = 10a.x = 0, y = 0b.No solutionc.x = 1, y = 1d.Infinite solutions

1. Break Down the Problem

We have a system of equations:

1. $2x + 3y = 5$ (Equation 1)
2. $4x + 6y = 10$ (Equation 2)

We need to solve this system using the matrix method and determine the nature of the solutions.

2. Relevant Concepts

The matrix form of the equations can be represented as $AX = B$, where: $A = \begin{pmatrix} 2 & 3 \\ 4 & 6 \end{pmatrix}, \quad X = \begin{pmatrix} x \\ y \end{pmatrix}, \quad B = \begin{pmatrix} 5 \\ 10 \end{pmatrix}$

3. Analysis and Detail

To analyze the system, we can compute the determinant of the coefficient matrix $A$: $\text{det}(A) = (2)(6) - (4)(3) = 12 - 12 = 0$ Since the determinant is zero, this indicates that the system might have either no solutions or infinitely many solutions.

Next, we need to check if the two equations are dependent by expressing the second equation in terms of the first.

Notice that if we multiply Equation 1 by 2: $2(2x + 3y) = 2(5) \implies 4x + 6y = 10$ We see that Equation 2 is just a scalar multiple of Equation 1, indicating that both equations represent the same line.

4. Verify and Summarize

Since both equations represent the same line, there are infinitely many solutions along that line. Therefore, this system does not yield a unique solution, confirming that:

Final Answer

d. Infinite solutions