Summary of Algebraic Expressions and Basic Geometry

## Introduction Algebra is the branch of mathematics that uses symbols and letters to represent numbers and relationships between them. Learning to construct and manipulate algebraic expressions helps you translate real-world situations into mathematical language and solve problems efficiently. > **Definition:** An algebraic expression is a combination of numbers, variables, and operations such as addition, subtraction, multiplication, division, and powers written in a single statement, for example $3x + 2$. ## Building Blocks of Algebraic Expressions ### Variables and Constants - **Variables** are symbols (usually letters) that represent unknown or changing values, e.g., $x$, $y$, $n$. - **Constants** are fixed values, e.g., $5$, $-3$. > **Definition:** A coefficient is the numerical factor multiplying a variable, for example in $4x$ the coefficient is $4$. ### Operations and Terms - A **term** is a part of an expression separated by $+$ or $-$, e.g., in $y+5$ the terms are $y$ and $5$. - Use parentheses to show grouping, e.g., $2(x+3)$ means multiply the sum $x+3$ by $2$. ## Translating Words into Expressions - Words describing change can be translated into expressions: - "add 1" becomes $+1$ so $y$ becomes $y+1$. - "take 1" or "subtract 1" becomes $-1$ so $y$ becomes $y-1$. - "add 5" becomes $+5$ so $y+5$. - "take 5" becomes $y-5$, etc. Example: "I add 8 counters to the bag, there are now $y+8$." This directly gives the expression $y+8$. > **Definition:** Equivalent expressions are expressions that have the same value for all values of the variables involved. ## Examples from the activity and how to form them - "I add 1 counter to the bag, there are now $y+1$." translates to $y+1$. - "I take 1 counter from the bag, which leaves $y-1$." translates to $y-1$. - "I add 5 counters to the bag, there are now $y+5$." translates to $y+5$. - "I take 5 counters from the bag, which leaves $y-5$." translates to $y-5$. - "I add 8 counters to the bag, there are now $y+8$." translates to $y+8$. - "I take 8 counters from the bag, which leaves $y-8$." translates to $y-8$. ### Translating numeric procedure descriptions - "I think of a number $n$. I halve the number then add $4$." Translate step-by-step: 1. Halve $n$ gives $\frac{1}{2}n$. 2. Add $4$ gives $\frac{1}{2}n + 4$. - "I think of a number $n$. I divide by $3$, then multiply by $2$." Step-by-step: 1. Divide by $3$ gives $\frac{1}{3}n$. 2. Multiply by $2$ gives $2\left(\frac{1}{3}n\right) = \frac{2}{3}n$. - "I think the expression is $\frac{2}{3}x + 4$." This matches the previous procedure if the final step includes adding $4$ after multiplying. - "I think the expression is $\frac{2}{3}x + 2$." This would mean after dividing by $3$ and multiplying by $2$ you then add $2$. ## Equivalent expressions and simplification - Combine like terms when possible: $3x + 2x = 5x$. - Use the distributive property: $a(b+c) = ab + ac$. Table: Common translations and their expressions | Phrase | Expression | | --- | --- | | add $1$ to $y$ | $y+1$ | | take $1$ from $y$ | $y-1$ | | add $5$ to $y$ | $y+5$ | | take $5$ from $y$ | $y-5$ | | add $8$ to $y$ | $y+8$ | | take $8$ from $y$ | $y-8$ | | halve $n$ then add $4$ | $\frac{1}{2}n + 4$ | | divide $n$ by $3$ then multiply by $2$ | $\frac{2}{3}n$ | ## Practical examples and real-world applications - Shopping: If an item costs $p$ dollars and you add a tax of $t$ dollars, the total is $p+t$. - Sharing: If $n$ people share $S$ items equally, each gets $\frac{S}{n}$. - Recipes: Doubling a recipe multiplies each ingredient by $2$, e.g., $2(\text{ingredient})$. Fun fact: Algebraic notation was standardized over centuries; the modern use of letters to represent unknowns became common in the 16th and 17th centuries. ## Practice problems 1. Translate: "I think of a number $x$. I add $3$, then multiply by $4$." Write the expression. 2. Simplify: $2x + 3x - 5$. 3. Are the expressions $2(\frac{1}{3}n)$ and $\frac{2}{3}n$ equivalent? E