Finding solutions to systems of linear equations can be done easily using something called Cramer's Rule. Let’s break it down step by step:

1. Set Up the System: Imagine you have a system like this: [ \begin{align*} 2x + 3y &= 8 \ 4x + y &= 10 \end{align*} ]

2. Form the Coefficient Matrix: Next, we make a matrix using the numbers in front of (x) and (y): [ A = \begin{pmatrix} 2 & 3 \ 4 & 1 \end{pmatrix} ]

3. Calculate the Determinant: Now, let’s find the determinant of this matrix: [ \text{det}(A) = (2)(1) - (3)(4) = 2 - 12 = -10 ]

4. Apply Cramer's Rule: To find the value of (x):

   • We replace the first column of the matrix with the numbers on the right side of the equations: [ A_x = \begin{pmatrix} 8 & 3 \ 10 & 1 \end{pmatrix} ]
   • Calculate the determinant for this new matrix: [ \text{det}(A_x) = (8)(1) - (3)(10) = 8 - 30 = -22 ]
   • Now, find (x): [ x = \frac{\text{det}(A_x)}{\text{det}(A)} = \frac{-22}{-10} = 2.2 ]

Now, you would repeat these steps for (y).

Using determinants makes it simpler to solve complicated systems of equations, showing how powerful they are in working with matrices!