To solve the system of linear equations:
[tex]\[
\begin{cases}
x + 3y = 8 \\
2x - y = 9
\end{cases}
\][/tex]
we can use the method of determinants, also known as Cramer's rule. Here is the step-by-step solution:
1. Identify the coefficients from the system of equations:
[tex]\[
\begin{cases}
a_1 = 1, \quad b_1 = 3, \quad c_1 = 8 \\
a_2 = 2, \quad b_2 = -1, \quad c_2 = 9
\end{cases}
\][/tex]
2. Calculate the determinant [tex]\(D\)[/tex] of the system:
[tex]\[
D = \begin{vmatrix}
a_1 & b_1 \\
a_2 & b_2
\end{vmatrix} = (1)(-1) - (2)(3) = -1 - 6 = -7
\][/tex]
Since [tex]\(D \neq 0\)[/tex], we know that the system of equations has a unique solution.
3. Calculate the determinant [tex]\(D_x\)[/tex] for the variable [tex]\(x\)[/tex]:
[tex]\[
D_x = \begin{vmatrix}
c_1 & b_1 \\
c_2 & b_2
\end{vmatrix} = (8)(-1) - (9)(3) = -8 - 27 = -35
\][/tex]
4. Calculate the determinant [tex]\(D_y\)[/tex] for the variable [tex]\(y\)[/tex]:
[tex]\[
D_y = \begin{vmatrix}
a_1 & c_1 \\
a_2 & c_2
\end{vmatrix} = (1)(9) - (2)(8) = 9 - 16 = -7
\][/tex]
5. Find the values of [tex]\(x\)[/tex] and [tex]\(y\)[/tex] using the determinants:
[tex]\[
x = \frac{D_x}{D} = \frac{-35}{-7} = 5
\][/tex]
[tex]\[
y = \frac{D_y}{D} = \frac{-7}{-7} = 1
\][/tex]
So, the solution to the system of equations is:
[tex]\[
x = 5, \quad y = 1
\][/tex]
