Responder
- a) \(x = 5\), \(y = 8\)
- b) \(x = 3\), \(y = 3\)
- c) \(x = 6\), \(y = 4\)
- d) \(x = 2\), \(y = -1\)
- e) \(x = 0\), \(y = -\frac{1}{2}\)
Solución
To solve the problems, we need to find the values of the variables \(x\) and \(y\) in each pair of sets \(A\) and \(B\) such that the sets are equal. This means that every element in set \(A\) must be equal to some element in set \(B\).
Let's analyze each case step by step.
### a) \( A=\{2 ; x ; 8\} \) e \( B=\{2 ; 5 ; y\} \)
1. From the sets, we can see that \(2\) is common in both sets.
2. For the sets to be equal, the remaining elements must also match:
- \(x\) must equal \(5\) (from set \(B\)).
- \(8\) must equal \(y\) (from set \(B\)).
Thus, we have:
- \(x = 5\)
- \(y = 8\)
### b) \( A=\{-1 ; 2x+3 ; 7\} \) e \( B=\{y-4 ; 9 ; 7\} \)
1. The common element is \(7\).
2. The remaining elements must match:
- \(-1\) must equal \(y - 4\) (from set \(B\)).
- \(2x + 3\) must equal \(9\) (from set \(B\)).
Now we can set up the equations:
1. \(y - 4 = -1\) → \(y = 3\)
2. \(2x + 3 = 9\) → \(2x = 6\) → \(x = 3\)
Thus, we have:
- \(x = 3\)
- \(y = 3\)
### c) \( A=\{3x-7 ; -5 ; 8y-1\} \) e \( B=\{11 ; -5 ; 10y-9\} \)
1. The common element is \(-5\).
2. The remaining elements must match:
- \(3x - 7 = 11\) (from set \(B\)).
- \(8y - 1 = 10y - 9\) (from set \(B\)).
Now we can set up the equations:
1. \(3x - 7 = 11\) → \(3x = 18\) → \(x = 6\)
2. \(8y - 1 = 10y - 9\) → \(2y = 8\) → \(y = 4\)
Thus, we have:
- \(x = 6\)
- \(y = 4\)
### d) \( A=\{x-y ; 5 ; 9\} \) e \( B=\{3 ; 3x+y ; 9\} \)
1. The common element is \(9\).
2. The remaining elements must match:
- \(x - y = 3\) (from set \(B\)).
- \(5 = 3x + y\) (from set \(B\)).
Now we can set up the equations:
1. \(x - y = 3\) → \(y = x - 3\)
2. Substituting \(y\) into the second equation:
\(5 = 3x + (x - 3)\) → \(5 = 4x - 3\) → \(4x = 8\) → \(x = 2\)
3. Now substituting \(x\) back to find \(y\):
\(y = 2 - 3 = -1\)
Thus, we have:
- \(x = 2\)
- \(y = -1\)
### e) \( A=\{-3, 3x-4y ; 1\} \) e \( B=\{-3 ; 2 ; 5x-2y\} \)
1. The common element is \(-3\).
2. The remaining elements must match:
- \(3x - 4y = 2\) (from set \(B\)).
- \(1 = 5x - 2y\) (from set \(B\)).
Now we can set up the equations:
1. \(3x - 4y = 2\) (Equation 1)
2. \(5x - 2y = 1\) (Equation 2)
We can solve these equations simultaneously.
Let's solve the system of equations:
\[
\begin{align*}
3x - 4y &= 2 \quad (1) \\
5x - 2y &= 1 \quad (2)
\end{align*}
\]
We can multiply Equation (1) by \(2\) and Equation (2) by \(4\) to eliminate \(y\):
\[
\begin{align*}
6x - 8y &= 4 \quad (3) \\
20x - 8y &= 4 \quad (4)
\end{align*}
\]
Now, subtract Equation (3) from Equation (4):
\[
20x - 6x = 4 - 4 \\
14x = 0 \\
x = 0
\]
Now substituting \(x = 0\) back into Equation (1):
\[
3(0) - 4y = 2 \\
-4y = 2 \\
y = -\frac{1}{2}
\]
Thus, we have:
- \(x = 0\)
- \(y = -\frac{1}{2}\)
### Summary of Results:
- a) \(x = 5\), \(y = 8\)
- b) \(x = 3\), \(y = 3\)
- c) \(x = 6\), \(y = 4\)
- d) \(x = 2\), \(y = -1\)
- e) \(x = 0\), \(y = -\frac{1}{2}\)
Respondido por UpStudy AI y revisado por un tutor profesional

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