To find the complex number \( x \) such that \( (3 - 4i) \cdot x = 25 \):
1. Let \( x = a + bi \), where \( a \) and \( b \) are real numbers.
2. Multiply the complex numbers:
[tex]\[
(3 - 4i)(a + bi)
\][/tex]
3. Use the distributive property:
[tex]\[
(3 - 4i)(a + bi) = 3a + 3bi - 4ai - 4i^2
\][/tex]
4. Recall that \( i^2 = -1 \):
[tex]\[
\Rightarrow 3a + 3bi - 4ai + 4b
\][/tex]
5. Separate the real and imaginary parts:
- Real part: \( 3a + 4b \)
- Imaginary part: \( 3b - 4a \)
6. Set up the system of linear equations:
[tex]\[
\begin{cases}
3a + 4b = 25 \\
3b - 4a = 0 \\
\end{cases}
\][/tex]
7. Solve the system of equations to find \( a \) and \( b \):
- From the second equation \( 3b - 4a = 0 \):
[tex]\[
b = \frac{4}{3}a
\][/tex]
- Substitute \( b \) in the first equation:
[tex]\[
3a + 4\left(\frac{4}{3}a\right) = 25
\][/tex]
[tex]\[
3a + \frac{16}{3}a = 25
\][/tex]
[tex]\[
9a + 16a = 75
\][/tex]
[tex]\[
25a = 75
\][/tex]
[tex]\[
a = 3
\][/tex]
- Substitute \( a = 3 \) back into \( b = \frac{4}{3}a \):
[tex]\[
b = \frac{4}{3} \cdot 3 = 4
\][/tex]
8. Thus, the complex number we are looking for is:
[tex]\[
x = 3 + 4i
\][/tex]
So, the correct answer is [tex]\( \boxed{3+4i} \)[/tex].
