Literal Equations - Algebra-1

1. Fundamental Concepts

Definition: Literal equations are equations that involve multiple variables, where the goal is to solve for one variable in terms of the others.
Isolation of Variables: The process of solving a literal equation involves isolating the desired variable on one side of the equation.
Properties of Equality: Use properties of equality (addition, subtraction, multiplication, division) to manipulate the equation and isolate the variable.

2. Key Concepts

Basic Rule: $$a + b = c \Rightarrow a = c - b$$

Proportional Relationships: $$\frac{a}{b} = \frac{c}{d} \Rightarrow a = \frac{bc}{d}$$

Application: Used in physics, chemistry, and engineering to rearrange formulas and solve for specific variables.

3. Examples

Example 1 (Basic)

Problem: Solve the equation $$d = rt$$ for $$t$$.

Step-by-Step Solution:

Start with the given equation: $$d = rt$$
Divide both sides by $$r$$: $$\frac{d}{r} = t$$
Rearrange to isolate $$t$$: $$t = \frac{d}{r}$$

Validation: Substitute $$d = 100$$, $$r = 50$$ → Original: $$100 = 50t \Rightarrow t = 2$$; Simplified: $$t = \frac{100}{50} = 2$$ ✓

Example 2 (Intermediate)

Problem: Solve the equation $$A = \pi r^2$$ for $$r$$.

Step-by-Step Solution:

Start with the given equation: $$A = \pi r^2$$
Divide both sides by $$\pi$$: $$\frac{A}{\pi} = r^2$$
Take the square root of both sides: $$r = \sqrt{\frac{A}{\pi}}$$

Validation: Substitute $$A = 100\pi$$ → Original: $$100\pi = \pi r^2 \Rightarrow r^2 = 100 \Rightarrow r = 10$$; Simplified: $$r = \sqrt{\frac{100\pi}{\pi}} = \sqrt{100} = 10$$ ✓

4. Problem-Solving Techniques

Isolate the Variable: Identify the variable you need to solve for and use algebraic operations to isolate it.
Use Inverse Operations: Apply inverse operations to both sides of the equation to maintain balance.
Check Your Work: Substitute the solved value back into the original equation to verify the solution.