Summary of Algebraic Modeling and Basic Geometry
Algebraic Modeling & Basic Geometry: Master Key Concepts
Introduction
Algebra word problems and linear functions connect real-life situations to mathematical expressions. In many everyday problems you can model cost, distance, or quantity with a linear function of the form $y=mx+b$. This material shows how to translate words into equations, solve them, and graph the resulting linear functions.
Definition: A linear function is a function of the form $y=mx+b$ where $m$ is the slope and $b$ is the $y$-intercept.
1. Translating word problems into linear equations
When a problem describes a fixed (one-time) cost and a per-item cost, you can write the total cost as a linear function. General template:
- Let $x$ be the number of items, tickets, pizzas, etc.
- Let $C$ be the total cost.
- If there is a per-item cost $p$ and a fixed fee $f$, then
$$C = p x + f$$
Example 1: Buying books online
Aron buys books. Each book costs $6$ and there is a fixed postage of $4$. Let $b$ be the number of books and $C$ the cost.
$$C = 6 b + 4$$
Evaluate:
- For $b=6$: $$C = 6\cdot 6 + 4 = 36 + 4 = 40$$
- For $b=12$: $$C = 6\cdot 12 + 4 = 72 + 4 = 76$$
- For $b=3$: $$C = 6\cdot 3 + 4 = 18 + 4 = 22$$
Example 2: Theatre tickets
Tickets cost $12$ each plus a booking fee of $3$. Let $t$ be the number of tickets and $C$ the cost.
$$C = 12 t + 3$$
Evaluate:
- For $t=6$: $$C = 12\cdot 6 + 3 = 72 + 3 = 75$$
- For $t=4$: $$C = 12\cdot 4 + 3 = 48 + 3 = 51$$
- For $t=10$: $$C = 12\cdot 10 + 3 = 120 + 3 = 123$$
Note: In the original notes one example evaluated $6$ tickets as $78$; check arithmetic carefully. The correct value is $75$ as shown above.
Example 3: Pizza delivery
A pizza party: each family-size pizza costs $24$ and delivery is a flat $5$. Let $p$ be the number of pizzas and $C$ the total cost.
$$C = 24 p + 5$$
Evaluate:
- For $p=8$: $$C = 24\cdot 8 + 5 = 192 + 5 = 197$$
- For $p=2$: $$C = 24\cdot 2 + 5 = 48 + 5 = 53$$
Note: The original notes gave $8$ pizzas cost $192$, which omitted the delivery fee. Include fixed fees unless the problem states otherwise.
Definition: The slope $m$ of a linear function $y=mx+b$ measures the rate of change: the change in $y$ when $x$ increases by $1$. The intercept $b$ is the value when $x=0$.
2. Perimeter and area problems leading to functions
Many geometry problems lead to linear expressions for perimeter and linear expressions for area in terms of one length.
Given shape problem
Suppose a composite shape has one side length called $L$ and the other sides add to a constant. The perimeter $P$ can often be written as a linear function of $L$ such as:
$$P = 2L + 22$$
This comes from summing two sides of length $L$ and other sides that total $22$.
If area $A$ is linear in $L$ for the given shape (for example, a rectangle of width $8$ and length $L$ plus additional constant-area parts), you might have:
$$A = 8L + 18$$
Always check the shape and dimensions to justify each term.
3. Graphing linear functions
A linear function $y=mx+b$ is a straight line. To graph it:
- Plot the $y$-intercept $(0,b)$.
- Use the slope $m$ as "rise over run" to find another point.
- Draw the line through the points.
Example: Graph $y=2x+3$
- $y$-intercept is $(0,3)$ since $b=3$.
- Slope $m=2$ means rise $2$ and run $1$. From $(0,3)$ go to $(1,5)$.
- Draw the straight line through these points.
4. Common translation phrases
- "Each" or "per" means multiply by the variable: $6$ per book $ o 6b$.
- "Plus a fixed fee" means add a constant: $+4$.
- "Total cost" or "C" is typically the dependent variable: $C = \text{(per-item)}\cdot x + \text{(fixed)}$.
| Phrase | Mathematical form | Example |
|---|---|---|
| Per item cost and fee | $C = p x + f$ | $C = 6b + 4$ |
| Only per item | $C = p x$ | $C = 12t$ if no fee |
| Fixed total | $C = f$ | One-time fee $f$ |
5. Checking and interpreting results
- Always substitute back to check
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Linear Word Problems
Klíčové pojmy: Translate word problems into $C=px+f$, Identify per-item cost as slope $p$ and fixed fee as intercept $f$, Evaluate linear functions by substitution, Graph $y=mx+b$ using intercept and slope, Check arithmetic and include all fees, Perimeter can be linear in one variable: $P=2L+22$ as example, Area can be linear in one variable: $A=8L+18$ as example, Use phrases like "each" => multiply, "plus" => add, Slope $m$ is rate of change, intercept $b$ is value at $x=0$, Verify units and interpret results