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If [tex]\( A=\left[\begin{array}{ll}
4 & 5 \\
8 & 7 \\
x & y
\end{array}\right], \quad B=\left[\begin{array}{ll}
0 & 2 \\
2 & 3 \\
4 & 5
\end{array}\right], \quad \text{and} \quad C=\left[\begin{array}{cc}
-1 & 2 \\
3 & -4 \\
0 & 1
\end{array}\right] \)[/tex]

and [tex]\((A+B)-C=\left[\begin{array}{cc} 5 & 4 \\ 7 & 14 \\ 7 & 6 \end{array}\right]\)[/tex],

then find the values of [tex]\( x \)[/tex] and [tex]\( y \)[/tex].

Sagot :

GinnyAnswer
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]
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