Which subtraction expression has the difference 1 + 4i

Subtraction Expression with Difference 1 + 4i

To find the subtraction expression that results in 1 + 4i, we need to solve each given expression step by step:

Expression 1:

$$(-2 + 6i) - (1 - 2i)$$ $$(-2 + 6i) - (1 - (-2i)) = -2 + 6i - 1 + 2i$$ $$= (-2 - 1) + (6i + 2i)$$ $$= -3 + 8i$$

This does not match 1 + 4i, so it’s not the correct expression.

Expression 2:

$(-2 + 6i) - (1 - 2i)$ (same as above, included twice by mistake)

Expression 3:

$$(3 + 5i) - (2 - i)$$ $$(3 + 5i) - (2 - i) = 3 + 5i - 2 + i$$ $$= (3 - 2) + (5i + i)$$ $$= 1 + 6i$$

This does not match 1 + 4i either.

Expression 4:

$$(3 + 5i) - (2 + i)$$ $$(3 + 5i) - (2 + i) = 3 + 5i - 2 - i$$ $$= (3 - 2) + (5i - i)$$ $$= 1 + 4i$$

This matches the required difference of 1 + 4i. Therefore, the correct subtraction expression is:

$(3 + 5i) - (2 + i)$

Explanation

Complex number subtraction involves taking the difference between two complex numbers. Recall that a complex number is in the form $a + bi$, where $a$ is the real part and $b$ is the imaginary part.

Subtracting Complex Numbers

The subtraction of two complex numbers $z_1 = a + bi$ and $z_2 = c + di$ is performed by individually subtracting their real and imaginary parts. Mathematically, this can be expressed as:

$$z_1 - z_2 = (a + bi) - (c + di)$$

Separation into Real and Imaginary Parts

To simplify this, we can separate the real and imaginary components:

$$z_1 - z_2 = (a - c) + (b - d)i$$

This means that the result of the subtraction will be another complex number where:

• The real part is $a - c$
• The imaginary part is $b - d$

Example

Let's consider an example to illustrate this.

Suppose we have two complex numbers $z_1 = 5 + 3i$ and $z_2 = 2 + 7i$. To find $z_1 - z_2$:

$$(z_1 - z_2) = (5 + 3i) - (2 + 7i)$$

Separate the real and imaginary parts:

$$(5 - 2) + (3 - 7)i = 3 - 4i$$

So, the result of subtracting $z_2$ from $z_1$ is:

$$z_1 - z_2 = 3 - 4i$$

This result shows a new complex number with a real part of 3 and an imaginary part of -4.

Key Points

• Separate real and imaginary parts: Treat them independently during subtraction.
• The result is always another complex number.
• Practice with examples to get comfortable with the operations.

Understanding this process is essential for working with complex numbers in any mathematical or engineering field.

Introduction

Combining like terms is a fundamental process in algebra that simplifies expressions. Like terms are terms that have the same variables raised to the same power, even if they have different coefficients.

Identifying Like Terms

To combine like terms, you first identify the terms that have the same variables and exponents. For example, in the expression:

$$3x^2 + 5x - 2x + 4 - x^2$$

The like terms are:

• $3x^2$ and $-x^2$ (both have the variable $x$ raised to the power of 2)
• $5x$ and $-2x$ (both have the variable $x$ raised to the power of 1)

Combining the Terms

1. Group the like terms together.
2. Add or subtract their coefficients.

For the expression given above, we can group and then combine like terms:

$$3x^2 - x^2 + 5x - 2x + 4$$

Combining the coefficients of the like terms, we get:

$$3x^2 - x^2 = 2x^2$$ $$5x - 2x = 3x$$

So, the simplified expression is:

$$2x^2 + 3x + 4$$

Special Cases

Constants, or terms without variables, are also considered like terms with each other. For instance, in the expression:

$$7 + 3x + 5 - 2x$$

The constants $7$ and $5$ can be combined:

$$7 + 5 = 12$$

Resulting in the simplified form:

$$3x - 2x + 12$$ $$x + 12$$

Benefits

Combining like terms:

• Simplifies expressions, making them easier to work with.
• Reduces complexity, especially useful in solving equations or inequalities.

Example

Let's look at another example:

$$4y + 7 - 5y + 3 + 2y$$

First, group the like terms:

$$(4y - 5y + 2y) + (7 + 3)$$

Now, combine the coefficients:

$$(4y - 5y + 2y) = (4 - 5 + 2)y = 1y = y$$ $$7 + 3 = 10$$

The simplified expression is:

$$y + 10$$

Being proficient at combining like terms is essential for simplifying algebraic expressions and solving algebra problems effectively.