Decide whether each statement is true or false. If false, correct the right side of the equation. (-2+7i) - (10-6i)= -12+i

Verified step by step guidance

1. Identify the problem: You need to subtract the complex numbers \((-2 + 7i)\) and \((10 - 6i)\) and then check if the result equals \(-12 + i\).
2. Recall the rule for subtracting complex numbers: Subtract the real parts and subtract the imaginary parts separately. That is, for \((a + bi) - (c + di)\), the result is \((a - c) + (b - d)i\).
3. Apply the subtraction to the given numbers: Calculate the real part as \(-2 - 10\) and the imaginary part as \$7 - (-6)$.
4. Write the result of the subtraction as a complex number: Combine the results from the previous step into the form \(x + yi\).
5. Compare your result with the given expression \(-12 + i\). If they are not equal, state that the original statement is false and provide the correct result.

Key Concepts

Complex Number Arithmetic

Complex numbers consist of a real part and an imaginary part, written as a + bi. Arithmetic operations like addition and subtraction are performed by combining like terms: real parts with real parts and imaginary parts with imaginary parts.

Subtraction of Complex Numbers

To subtract complex numbers, subtract the real parts and the imaginary parts separately. For example, (a + bi) - (c + di) = (a - c) + (b - d)i, ensuring careful sign handling to avoid errors.

Verification of Equations Involving Complex Numbers

To verify if an equation with complex numbers is true, perform the indicated operations and compare both sides. If they differ, identify and correct the mistake by recalculating the real and imaginary parts accurately.