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Linear Algebra 2
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1. Equation: (2x + 3i) + (x - 2i) = 5 + iWorking Out:
Combine like terms:
3x + i = 5 + i
Subtract i from both sides to isolate the variable:
3x = 5
Divide both sides by 3 to find the value of x:
x=5/3
, 2. Equation: (3 + 4i)x = 9 - 6i
Working Out:
Divide both sides by (3 + 4i) to isolate the variable:
x = (9 - 6i) / (3 + 4i)
x = (9 - 6i)(3 - 4i) / (3 + 4i)(3 - 4i)
x = (27 - 36i - 18i + 24i^2) / (9 - 16i^2)
x = (27 - 54i - 24) / (9 + 16)
x = (3 - 6i) / 5
x = 3/5 - (6/5)i
3. Equation: |2x - 5i| = 3
Working Out:
For the absolute value to be 3, the expression inside the absolute value must be ±3:
2x - 5i = 3 or 2x - 5i = -3
Solve for x in both cases:
2x = 3 + 5i or 2x = -3 + 5i
x = (3 + 5i) / 2 or x = (-3 + 5i) / 2
4. Equation: (x + 2i)(x - 2i) = 12
Working Out:
Use the difference of squares formula to simplify the left side:
(x + 2i)(x - 2i) = x^2 - (2i)^2
(x + 2i)(x - 2i) = x^2 - 4i^2
(x + 2i)(x - 2i) = x^2 + 4
Now set the equation equal to 12:
x^2 + 4 = 12
Subtract 4 from both sides to isolate the variable:
x^2 = 8
Take the square root of both sides to find the two solutions for x:
x = ±√8
x = ±2√2
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