Sure, let's solve for [tex]\( x \)[/tex] and [tex]\( y \)[/tex] step-by-step given the matrices [tex]\( A \)[/tex], [tex]\( B \)[/tex], and [tex]\( C \)[/tex], and the resulting matrix [tex]\((A + B) - C\)[/tex].
Given:
[tex]\[ A = \left[\begin{array}{cc} 4 & 5 \\ 8 & 7 \\ x & y \end{array}\right], \quad B = \left[\begin{array}{cc} 0 & 2 \\ 2 & 3 \\ 4 & 5 \end{array}\right], \quad C = \left[\begin{array}{cc} -1 & 2 \\ 3 & -4 \\ 0 & 1 \end{array}\right] \][/tex]
And the result is:
[tex]\[ (A + B) - C = \left[\begin{array}{cc} 5 & 4 \\ 7 & 14 \\ 7 & 6 \end{array}\right] \][/tex]
### Step 1: Calculate [tex]\( A + B \)[/tex]
Let's add matrices [tex]\( A \)[/tex] and [tex]\( B \)[/tex]:
[tex]\[ A + B = \left[\begin{array}{cc} 4 + 0 & 5 + 2 \\ 8 + 2 & 7 + 3 \\ x + 4 & y + 5 \end{array}\right] = \left[\begin{array}{cc} 4 & 7 \\ 10 & 10 \\ x + 4 & y + 5 \end{array}\right] \][/tex]
### Step 2: Calculate [tex]\( (A + B) - C \)[/tex]
Now, let's subtract matrix [tex]\( C \)[/tex] from [tex]\( A + B \)[/tex]:
[tex]\[ (A + B) - C = \left[\begin{array}{cc} 4 & 7 \\ 10 & 10 \\ x + 4 & y + 5 \end{array}\right] - \left[\begin{array}{cc} -1 & 2 \\ 3 & -4 \\ 0 & 1 \end{array}\right] \][/tex]
[tex]\[ (A + B) - C = \left[\begin{array}{cc} 4 - (-1) & 7 - 2 \\ 10 - 3 & 10 - (-4) \\ (x + 4) - 0 & (y + 5) - 1 \end{array}\right] = \left[\begin{array}{cc} 5 & 5 \\ 7 & 14 \\ x + 4 & y + 4 \end{array}\right] \][/tex]
### Step 3: Set up equations and solve for [tex]\( x \)[/tex] and [tex]\( y \)[/tex]
We know that:
[tex]\[ (A + B) - C = \left[\begin{array}{cc} 5 & 4 \\ 7 & 14 \\ 7 & 6 \end{array}\right] \][/tex]
By comparing the third row:
[tex]\[ [x + 4, y + 4] = [7, 6] \][/tex]
This gives us two equations:
1. [tex]\( x + 4 = 7 \)[/tex]
2. [tex]\( y + 4 = 6 \)[/tex]
Solving these equations:
1. [tex]\( x + 4 = 7 \)[/tex]
[tex]\[ x = 7 - 4 \][/tex]
[tex]\[ x = 3 \][/tex]
2. [tex]\( y + 4 = 6 \)[/tex]
[tex]\[ y = 6 - 4 \][/tex]
[tex]\[ y = 2 \][/tex]
Thus, the values for [tex]\( x \)[/tex] and [tex]\( y \)[/tex] are:
[tex]\[ x = 3 \][/tex]
[tex]\[ y = 2 \][/tex]
