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7. If [tex]\( x, y \in \mathbb{R} \)[/tex] and [tex]\( (5y - 2) + i(3x - y) = 3 - 7i \)[/tex], find the values of [tex]\( x \)[/tex] and [tex]\( y \)[/tex].

Sagot :

GinnyAnswer
To solve for \(x\) and \(y\) given the equation \((5y - 2) + i(3x - y) = 3 - 7i\), we need to equate the real and imaginary parts separately. Let's break it down step-by-step:

1. Identify the real and imaginary parts:
[tex]\[ (5y - 2) + i(3x - y) = 3 - 7i \][/tex]

2. Equate the real parts:
[tex]\[ 5y - 2 = 3 \][/tex]
Solve for \(y\):
[tex]\[ 5y - 2 = 3 \\ 5y = 5 \\ y = 1 \][/tex]

3. Equate the imaginary parts:
[tex]\[ 3x - y = -7 \][/tex]
Substitute \(y = 1\):
[tex]\[ 3x - 1 = -7 \\ 3x = -6 \\ x = -2 \][/tex]

Hence, the values of \(x\) and \(y\) that satisfy the equation \((5y - 2) + i(3x - y) = 3 - 7i\) are:
[tex]\[ x = -2 \quad \text{and} \quad y = 1 \][/tex]
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