Sure, let's solve the given system. We have a matrix equation, which we will break down into individual equations to solve for [tex]\(x\)[/tex], [tex]\(y\)[/tex], and [tex]\(z\)[/tex].
The matrix equation given is:
[tex]\[
\left(\begin{array}{cc}
x + y & z - x \\
y + 2z & x
\end{array}\right) = \left(\begin{array}{cc}
3 & 2 \\
8 & 1
\end{array}\right)
\][/tex]
By comparing the corresponding elements in the two matrices, we can set up the following equations:
1. For the element in the first row, first column:
[tex]\[x + y = 3\][/tex]
2. For the element in the first row, second column:
[tex]\[z - x = 2\][/tex]
3. For the element in the second row, first column:
[tex]\[y + 2z = 8\][/tex]
4. For the element in the second row, second column:
[tex]\[x = 1\][/tex]
Now let's solve these step by step.
Step 1: From equation 4, we have:
[tex]\[x = 1\][/tex]
Step 2: Substitute [tex]\(x = 1\)[/tex] into equation 1:
[tex]\[1 + y = 3\][/tex]
[tex]\[y = 3 - 1 = 2\][/tex]
Step 3: Substitute [tex]\(x = 1\)[/tex] into equation 2:
[tex]\[z - 1 = 2\][/tex]
[tex]\[z = 2 + 1 = 3\][/tex]
Step 4: Finally, let's verify the value of [tex]\(y\)[/tex] with equation 3:
[tex]\[2 + 2 \cdot 3 = 8\][/tex]
[tex]\[2 + 6 = 8\][/tex]
[tex]\(8 = 8\)[/tex]
The values satisfy all the given equations. Therefore, the values of the variables are:
[tex]\[
x = 1, \quad y = 2, \quad z = 3
\][/tex]
Thus, we have solved for [tex]\(x\)[/tex], [tex]\(y\)[/tex], and [tex]\(z\)[/tex].
