Study Notes: Key Concepts in College Algebra and Trigonometry

Linear Equations and Inequalities

Solving Linear Equations

Linear equations are equations of the first degree, meaning the variable is raised only to the first power. Solving such equations involves isolating the variable on one side of the equation.

• Standard Form: $ax + b = c$

• Solution Steps:

  1. Combine like terms on each side if necessary.

  2. Isolate the variable by adding or subtracting terms.

  3. Divide or multiply to solve for the variable.

• Example: Solve $5x - 7 = 2x + 8$ Solution:

  • Subtract $2x$ from both sides: $3x - 7 = 8$

  • Add $7$to both sides: $3x = 15$

  • Divide by $3$:$x = 5$

Solving and Expressing Inequalities

Inequalities are mathematical statements that compare expressions using inequality symbols (<, >, ≤, ≥). Solutions are often expressed in interval notation.

• Interval Notation: A way to describe the set of solutions using parentheses and brackets.

• Example: Solve and express in interval notation: $-3 < 4x + 1 < 9$ Solution:

  • Subtract $1$:$-4 < 4x < 8$

  • Divide by $4$:$-1 < x < 2$

  • Interval Notation: $(-1, 2)$

Functions and Their Properties

Domain and Range

The domain of a function is the set of all possible input values (x-values), and the range is the set of all possible output values (y-values).

• Example: For the relation {(3,3), (1,-1), (0,-3), (a,b)}:

  • Domain: {3, 1, 0, a}

  • Range: {3, -1, -3, b}

Finding x- and y-Intercepts

The x-intercept is where the graph crosses the x-axis ($y=0$), and the y-intercept is where it crosses the y-axis ($x=0$).

• Example: For $2x + 3y = 6$:

  • x-intercept: Set $y=0$: $2x=6 \Rightarrow x=3$

  • y-intercept: Set $x=0$: $3y=6 \Rightarrow y=2$

Slope and Equation of a Line

The slope of a line measures its steepness and is calculated as $m = \frac{y_2 - y_1}{x_2 - x_1}$. The equation of a line in slope-intercept form is $y = mx + b$.

• Parallel Lines: Have the same slope.

• Example: Find the equation of the line parallel to $3x - y = 5$ that passes through $(-1,4)$.

  • Rewrite $3x - y = 5$ as $y = 3x - 5$; slope $m=3$.

  • Use point-slope form: $y - 4 = 3(x + 1)$

  • Simplify: $y = 3x + 7$

Systems of Equations

Solving Systems of Linear Equations

A system of equations is a set of two or more equations with the same variables. Solutions are values that satisfy all equations simultaneously.

• Methods: Substitution, elimination, or graphing.

• Example: Solve $2x - 3y = 1$ and $x + 4y = 5$.

  • Solve one equation for $x$ or $y$ and substitute into the other.

Applications and Word Problems

Investment Problems

These problems involve dividing an amount of money between accounts with different interest rates.

• Example: $10,000$ is split between two accounts: one at $5\%$ and one at $6\%$. If the total interest in a year is $560$, how much is in each account?

  • Let $x$ = amount at $5\%$, $y$ = amount at $6\%$.

  • $x + y = 10,000$

  • $0.05x + 0.06y = 560$

  • Solve the system for $x$ and $y$.

Mixture and Sales Problems

These involve combining items or prices to meet certain conditions.

• Example: A store sells apples for $1.00$ and oranges for $2.00$. A customer pays $14$for $10$ items. Find the number of each item.

  • Let $x$ = apples, $y$ = oranges.

  • $x + y = 10$

  • $1.00x + 2.00y = 14$

  • Solve the system for $x$ and $y$.

Geometry Applications

Some problems involve geometric figures, such as rectangles, and require setting up equations based on perimeter or area formulas.

• Perimeter of a Rectangle: $P = 2(l + w)$

• Example: The length of a rectangle is $3$meters less than twice the width. The perimeter is $18$ meters. Find the width.

  • Let $w$ = width, $l = 2w - 3$

  • $2(l + w) = 18$

  • Substitute and solve for $w$.

Function Operations and Composition

Function Operations

Functions can be added, subtracted, multiplied, or divided to create new functions.

• Example: Given $f(x) = x^2 - x$ and $g(x) = x + 4$, find $f - g$ and $g(1)$.

  • $f - g = f(x) - g(x) = (x^2 - x) - (x + 4) = x^2 - 2x - 4$

  • $g(1) = 1 + 4 = 5$

Graphing Linear Functions

To graph a linear function, identify the slope and y-intercept, plot the intercept, and use the slope to find another point.

• Example: Graph $2x + 5 = 9$.

  • Solve for $y$ if necessary, plot intercepts, and draw the line.

Summary Table: Key Concepts

Concept	Key Formula/Method	Example
Linear Equation	$ax + b = c$	$5x - 7 = 2x + 8$
System of Equations	Substitution/Elimination	$2x - 3y = 1$, $x + 4y = 5$
Domain/Range	List or interval notation	Domain: {3, 1, 0, a}
Perimeter of Rectangle	$P = 2(l + w)$	$l = 2w - 3$, $P = 18$
Function Operations	$(f - g)(x) = f(x) - g(x)$	$f(x) = x^2 - x$, $g(x) = x + 4$