To solve a system of linear equations using Cramer's Rule, we use determinants to find the values of the unknowns. Let's go through the steps required for the given system of linear equations:
[tex]\[
\begin{array}{l}
10 x - y = 3 \\
5 x - 2 y = -24
\end{array}
\][/tex]
1. Step 1: Write the coefficient matrix [tex]\( A \)[/tex]
The coefficient matrix [tex]\( A \)[/tex] for the given system is:
[tex]\[
A = \begin{pmatrix}
10 & -1 \\
5 & -2
\end{pmatrix}
\][/tex]
2. Step 2: Calculate the determinant of [tex]\( A \)[/tex], denoted as [tex]\( \det(A) \)[/tex]
[tex]\[
\det(A) = \begin{vmatrix}
10 & -1 \\
5 & -2
\end{vmatrix}
\][/tex]
3. Finding the determinant [tex]\( \det(A) \)[/tex]
[tex]\[
\det(A) = (10 \times -2) - (-1 \times 5) = -20 + 5 = -15
\][/tex]
So, the determinant of [tex]\( A \)[/tex] is [tex]\( -15 \)[/tex].
4. Step 3: Replace the columns of [tex]\( A \)[/tex] with the constants column [tex]\([3, -24]^T\)[/tex] to solve for each variable
- To solve for [tex]\( x \)[/tex], replace the first column of [tex]\( A \)[/tex] with the constants column:
[tex]\[
A_x = \begin{pmatrix}
3 & -1 \\
-24 & -2
\end{pmatrix}
\][/tex]
Then calculate [tex]\( \det(A_x) \)[/tex].
- To solve for [tex]\( y \)[/tex], replace the second column of [tex]\( A \)[/tex] with the constants column:
[tex]\[
A_y = \begin{pmatrix}
10 & 3 \\
5 & -24
\end{pmatrix}
\][/tex]
Then calculate [tex]\( \det(A_y) \)[/tex].
5. Step 4: Calculate the determinants [tex]\( \det(A_x) \)[/tex] and [tex]\( \det(A_y) \)[/tex]
- For [tex]\( \det(A_x) \)[/tex]:
[tex]\[
\det(A_x) = \begin{vmatrix}
3 & -1 \\
-24 & -2
\end{vmatrix} = (3 \times -2) - (-1 \times -24) = -6 - 24 = -30
\][/tex]
- For [tex]\( \det(A_y) \)[/tex]:
[tex]\[
\det(A_y) = \begin{vmatrix}
10 & 3 \\
5 & -24
\end{vmatrix} = (10 \times -24) - (3 \times 5) = -240 - 15 = -255
\][/tex]
6. Step 5: Solve for [tex]\( x \)[/tex] and [tex]\( y \)[/tex] using Cramer's Rule:
[tex]\[
x = \frac{\det(A_x)}{\det(A)} = \frac{-30}{-15} = 2
\][/tex]
[tex]\[
y = \frac{\det(A_y)}{\det(A)} = \frac{-255}{-15} = 17
\][/tex]
From the steps outlined:
- The first determinant calculated is [tex]\(\det(A)\)[/tex].
- The second determinant calculated is [tex]\(\det(A_x)\)[/tex].
- The third determinant calculated is [tex]\(\det(A_y)\)[/tex].
Hence, the minimum number of determinants needed to solve for all unknowns in this system using Cramer's Rule is 3.
