Linear Equations and Number Sets: College Algebra Study Notes

Linear Equations

Solving Linear Equations

Linear equations are algebraic equations in which each term is either a constant or the product of a constant and a single variable. Solving a linear equation involves finding the value of the variable that makes the equation true.

• Definition: A linear equation in one variable has the general form $ax + b = c$, where $a$, $b$, and $c$ are constants.

• Steps to Solve:

  1. Distribute any multiplication over parentheses.

  2. Combine like terms on each side of the equation.

  3. Isolate the variable on one side by adding or subtracting terms.

  4. Solve for the variable by dividing both sides by the coefficient.

• Example: Solve $-3(6x - 17) + 18x = 8x + 51 + 3x$

  • Distribute: $-3(6x - 17) = -18x + 51$

  • Rewrite: $-18x + 51 + 18x = 8x + 51 + 3x$

  • Combine like terms: $51 = 11x + 51$

  • Subtract 51 from both sides: $0 = 11x$

  • Divide by 11: $x = 0$

Key Formula:

$ax + b = c \implies x = \frac{c - b}{a}$

Number Sets and Irrational Numbers

Classifying Numbers: Rational vs. Irrational

Numbers can be classified into different sets based on their properties. Two important categories in algebra are rational and irrational numbers.

• Rational Numbers: Numbers that can be expressed as the quotient of two integers, i.e., $\frac{p}{q}$ where $p$ and $q$ are integers and $q \neq 0$.

• Irrational Numbers: Numbers that cannot be written as a simple fraction. Their decimal expansions are non-repeating and non-terminating.

• Examples of Irrational Numbers:

  • $\sqrt{24}$ (since 24 is not a perfect square)

  • $9\pi$ (since $\pi$ is irrational, any nonzero multiple of $\pi$ is also irrational)

• Examples of Rational Numbers:

  • $-1028$ (an integer)

  • $-2050/5$ (simplifies to $-410$, an integer)

  • $-3/25$ (a fraction of integers)

  • $0$ (an integer)

  • $4/21$ (a fraction of integers)

  • $1$ (an integer)

  • $11$ (an integer)

Key Point: Irrational numbers cannot be written as exact fractions and include roots of non-perfect squares and multiples of $\pi$.

Table: Classification of Elements in Set M

The following table classifies the elements of set M as rational or irrational:

Element	Classification
-1028	Rational
-2050/5	Rational
-3/25	Rational
0	Rational
4/21	Rational
1	Rational
11	Rational
$\sqrt{24}$	Irrational
$9\pi$	Irrational

Example: $\sqrt{24}$ is irrational because 24 is not a perfect square. $9\pi$ is irrational because $\pi$ is irrational.

Additional info: The problems provided are typical of College Algebra, focusing on solving linear equations and classifying numbers within sets.