Learning Objectives
In this section students will:
- 1.1.1 – Evaluate algebraic expressions
- 1.1.2 – Add and subtract algebraic expressions
- 1.1.3 – Multiply algebraic expressions
- 1.1.4 – Perform operations with algebraic expressions of several variables
- 1.1.5 – Test possible solutions to algebraic equations
- 1.1.6 – Translate words into algebraic expressions
Elaina, Ira, and Halle are visiting New York City, and all three will use the subway to get around. Elaina plans to take 20 rides on the subway. Since a new MetroCard costs $1, and each ride costs $2.75, the equation below shows that Elaina can expect to pay $56.
[latex]$$ \underbrace{2.75}_{\substack{\text{cost}\\\text{per ride}}}\times\underbrace{20}_{\substack{\text{\# of}\\\text{rides}}}+\underbrace{1}_{\substack{\text{MetroCard}\\\text{cost}}}=56 $$[/latex]
If Ira plans to take 15 rides on the subway, the equation [latex]2.75\times 15+1=42.25[/latex] shows that he can expect to pay $42.25. Similarly, Halle’s 12 rides will cost her [latex]2.75\times 12+1=34[/latex] dollars.
At this point, we’ve done three different calculations, but the only quantity that changed in each was the number of rides. To represent all of these calculations in one line, we can let the letter r represent the number of rides and write
[latex]$$ 2.75\times r+1 $$[/latex]
This is an example of an algebraic expression, which is a powerful idea in math that allows us to represent many different possible calculations in one line. The expression [latex]2.75r+1[/latex] can represent the cost to ride the subway any number of times, and by replacing r with any number, we can choose to focus on one particular calculation.
Since algebraic expressions represent quantities, we can add, subtract, and multiply them like we can with other quantities. Also, since setting one quantity equal to another creates an equation, we can make equations with algebraic expressions. Algebraic expressions and equations are the building blocks of algebra, so this section will help us grow more familiar with them.
1.1.1 – Evaluating Algebraic Expressions
In mathematics, we may see expressions such as [latex]x+5,\; \frac{4}{3}\pi {r}^{3},\;[/latex] or [latex]\sqrt{2{m}^{3}{n}^{2}}.[/latex] In the expression [latex]x+5,[/latex] the number [latex]5[/latex] is called a constant because it does not vary; it always has a value of 5. On the other hand, we call x a variable because its value may change. An algebraic expression is a collection of constants and variables joined together by the algebraic operations of addition, subtraction, multiplication, and division.
You are probably familiar with exponents like [latex]2^6[/latex] or [latex]\left(-4\right)^3[/latex]. Exponents provide a quick way to say that we are multiplying by the same number several times. The base of the exponent says which number we are multiplying by and the exponent says how many times. For instance, [latex]2^6 = 2\cdot 2\cdot 2\cdot 2\cdot 2\cdot 2[/latex] means that we