Equations and Linear Equations: Basic Terminology and Methods

Basic Terminology of Equations

Definition of an Equation

An equation is a mathematical statement that asserts the equality of two expressions. Equations are fundamental in algebra and are used to represent relationships between variables and constants.

• Equation: A statement that two expressions are equal.

• Solution: Any number that makes an equation true when substituted for the variable.

To solve an equation means to find all values of the variable that make the equation true. This is typically done by isolating the variable using algebraic operations. Any operation performed on one side of the equation must also be performed on the other side to maintain equality.

• We can perform the same operation on both sides of an equation to simplify or solve it.

Solving Linear Equations

Linear Equations in One Variable

A linear equation in one variable is an equation that can be written in the form:

$ax + b = 0$

• Where a and b are real numbers, and a \neq 0.

• Linear equations are also called first-degree equations because the highest power of the variable is 1.

Examples of linear equations:

• $2x + 3 = 7$

• $x - 5 = 0$

Examples of nonlinear equations:

• $x^2 + 3x = 7$

• $2^x = 8$

Solving Linear Equations: Steps

• Combine like terms on each side of the equation.

• Use addition or subtraction to get all variable terms on one side and constants on the other.

• Use multiplication or division to isolate the variable.

• Check your solution by substituting it back into the original equation.

Practice: Solving Linear Equations

• Example 1: Solve $6(3x - 1) = 8 - (10x - 14)$

• Example 2: Solve $0.04(x - 12) + 0.06x = 1.52$

• Example 3: Solve $\dfrac{2x + 4}{3} + \dfrac{1}{2}x = \dfrac{1}{4}x - \dfrac{7}{3}$

• Example 4: Solve $\dfrac{1}{14}(3x - 2) = \dfrac{x + 10}{10}$

Types of Equations

• Identity: An equation that is true for all values of the variable.

  • Both sides can be made to look identical after simplification.

  • If the variable terms cancel and a true statement remains (e.g., $0 = 0$), the equation is an identity.

  • The solution set is all real numbers.

• Conditional Equation: True for some values of the variable, but not all.

  • Typical equations with a finite number of solutions.

  • Solving yields a specific solution set.

• Contradiction: False for any value of the variable.

  • Variable terms cancel and a false statement remains (e.g., $0 = 5$).

  • The solution set is empty (no solution).

Example: Identifying Types of Equations

• a. $-2(x + 4) + 3x = x - 8$

• b. $5x - 4 = 11$

• Determine if each is an identity, a conditional equation, or a contradiction, and give the solution set.

Solving for a Specified Variable (Literal Equations)

Literal Equations

A literal equation is an equation that involves two or more variables. These equations are often formulas from geometry, physics, or other sciences. Solving a literal equation means isolating one variable in terms of the others.

• We use algebraic techniques to solve for the specified variable.

Examples: Solving for a Specified Variable

• a. $I = Prt$, solve for $t$

  • Divide both sides by $Pr$:

  • $t = \dfrac{I}{Pr}$

• b. $A - P = Prt$, solve for $P$

  • Add $P$ to both sides: $A = P + Prt$

  • Factor $P$: $A = P(1 + rt)$

  • Divide both sides by $(1 + rt)$:

  • $P = \dfrac{A}{1 + rt}$

• c. $3(2x - 5a) + 4b = 4x - 2$, solve for $x$

  • Expand: $6x - 15a + 4b = 4x - 2$

  • Subtract $4x$ from both sides: $2x - 15a + 4b = -2$

  • Add $15a - 4b$ to both sides: $2x = -2 + 15a - 4b$

  • Divide by 2: $x = \dfrac{-2 + 15a - 4b}{2}$

• d. $S = 2lw + 2wh + 2hl$, solve for $h$

  • Subtract $2lw$ from both sides: $S - 2lw = 2wh + 2hl$

  • Factor $h$: $S - 2lw = 2h(w + l)$

  • Divide both sides by $2(w + l)$:

  • $h = \dfrac{S - 2lw}{2(w + l)}$

Summary Table: Types of Equations

Type	Description	Solution Set	Example
Identity	True for all values of the variable	All real numbers	$2(x + 1) = 2x + 2$
Conditional	True for some values of the variable	Specific value(s)	$x + 3 = 7$
Contradiction	False for all values of the variable	No solution	$x + 2 = x + 5$

Additional info: The above notes expand on the provided worksheet by filling in standard definitions, solution steps, and examples for clarity and completeness, as expected in a college algebra context.