Precalculus Study Guide: Linear Equations and Complex Numbers

Solving Linear Equations

Linear Expressions vs. Linear Equations

A linear expression is an algebraic expression of the form ax + b, where a and b are constants. A linear equation is a statement that two linear expressions are equal, typically written as ax + b = c.

• Linear Expression Example: $2x + 3$

• Linear Equation Example: $2x + 3 = 7$

• Solving for a known x: Substitute the value of x and simplify.

• Solving for an unknown x: Find the value(s) of x that make the equation true.

Steps to Solve Linear Equations

1. Distribute constants: Apply the distributive property to remove parentheses.

2. Combine like terms: Add or subtract terms with the same variable.

3. Group terms: Move all terms with x to one side and constants to the other.

4. Isolate x: Solve for x by performing inverse operations.

5. Check solution: Substitute the value of x back into the original equation.

Example

• Equation: $3(2 - 5x) = 4x + 25$

• Step 1: Distribute: $6 - 15x = 4x + 25$

• Step 2: Group terms: $6 - 25 = 15x + 4x$

• Step 3: Combine like terms and solve for x.

Linear Equations with Fractions

When linear equations contain fractions, use the Least Common Denominator (LCD) to eliminate fractions before solving.

• Multiply both sides by the LCD to clear denominators.

• Proceed with the standard steps for solving linear equations.

Example

• Equation: $\frac{1}{2}(x + 2) - \frac{1}{3}x = \frac{1}{12}$

• Step 1: LCD is 12. Multiply both sides by 12 to clear denominators.

• Step 2: Solve the resulting linear equation.

Categorizing Linear Equations

Linear equations can be classified based on the number of solutions:

• Conditional Equation: Has exactly one solution.

• Identity: True for all real numbers (infinitely many solutions).

• Inconsistent Equation: Has no solution.

To determine the type, solve the equation and analyze the result.

Rational Equations

Definition and Solution

A rational equation is an equation that contains one or more rational expressions (fractions with variables in the denominator).

• To solve, multiply both sides by the LCD to eliminate denominators.

• Check for extraneous solutions: Solutions that make any denominator zero are not valid.

Example

• Equation: $\frac{x}{x-1} = \frac{7}{6}$

• Restriction: $x \neq 1$ (since denominator cannot be zero)

• Step 1: Cross-multiply and solve for x.

• Step 2: Check that the solution does not violate the restriction.

Square Roots of Negative Numbers and Imaginary Unit

Square Roots of Negative Numbers

• The square root of a positive number is real, but the square root of a negative number is not real.

• To handle this, mathematicians defined the imaginary unit $i$ such that $i = \sqrt{-1}$.

• Any square root of a negative number can be written as $\sqrt{-b} = i\sqrt{b}$, where $b > 0$.

Examples

• $\sqrt{-4} = 2i$

• $\sqrt{-17} = i\sqrt{17}$

Powers of the Imaginary Unit

Properties of $i$

• $i^1 = i$

• $i^2 = -1$

• $i^3 = -i$

• $i^4 = 1$

• These powers repeat every four exponents.

Shortcut for Evaluating Powers of $i$

• Divide the exponent by 4 and use the remainder to determine the value:

Remainder	Value
0	1
1	i
2	-1
3	-i

Complex Numbers

Definition and Standard Form

A complex number is a number of the form $a + bi$, where a and b are real numbers and $i$ is the imaginary unit.

• Real part: $a$

• Imaginary part: $b$

Examples

• $4 - 3i$: $a = 4$, $b = -3$

• $0 + 7i$: $a = 0$, $b = 7$

Adding and Subtracting Complex Numbers

• Add or subtract the real parts and the imaginary parts separately.

• Express the answer in standard form $a + bi$.

Example: $(4 + 8i) + (2 + 3i) = (4 + 2) + (8i + 3i) = 6 + 11i$

Multiplying Complex Numbers

• Use the distributive property (FOIL) as with binomials.

• Replace $i^2$ with $-1$ and combine like terms.

Example: $(3i)(7 - 2i) = 21i - 6i^2 = 21i - 6(-1) = 21i + 6$

Complex Conjugates

• The conjugate of $a + bi$ is $a - bi$.

• Multiplying a complex number by its conjugate yields a real number: $(a + bi)(a - bi) = a^2 + b^2$

Dividing Complex Numbers

• To divide by a complex number, multiply numerator and denominator by the conjugate of the denominator to make the denominator real.

• Expand and simplify to standard form.

Example: $\frac{3}{1 + 2i}$ Multiply numerator and denominator by $1 - 2i$.

Summary Table: Operations with Complex Numbers

Operation	Method	Example
Add/Subtract	Combine real and imaginary parts	$(3 + 2i) + (1 - 5i) = 4 - 3i$
Multiply	FOIL, replace $i^2$ with $-1$	$(2 + i)(3 - 4i) = 6 - 8i + 3i - 4i^2 = 6 - 5i + 4 = 10 - 5i$
Divide	Multiply by conjugate of denominator	$\frac{1}{2 + i} \times \frac{2 - i}{2 - i}$

Key Formulas

• Standard form of a complex number: $a + bi$

• Conjugate: $a - bi$

• Multiplying conjugates: $(a + bi)(a - bi) = a^2 + b^2$

• Powers of $i$: $i^1 = i$, $i^2 = -1$, $i^3 = -i$, $i^4 = 1$ (repeat every 4)

Additional info: These notes cover foundational Precalculus topics including linear equations, rational equations, and complex numbers, which are essential for further study in algebra, trigonometry, and calculus.