Lesson 2: Algebraic Expressions

New Concept

Learn to translate everyday language into powerful algebraic expressions. We'll build expressions with variables and operations, then evaluate them using given values or common formulas to find real-world answers for distance, profit, and more.

What’s next

Next, you'll tackle interactive examples on writing expressions from word problems, then move on to evaluating them and applying common algebraic formulas.

Property

An algebraic expression, or simply an expression, is any meaningful combination of numbers, variables, and operation symbols.

To write an algebraic expression:

1. Identify the unknown quantity and write a short phrase to describe it.
2. Choose a variable to represent the unknown quantity.
3. Use mathematical symbols to represent the relationship.

Examples

• The phrase "a number x increased by 12" translates to the expression $x + 12$.
• To represent "8 times the price p", you write the expression $8p$.
• "The total cost C split among 4 friends" is written as the expression $\frac{C}{4}$.

Property

When we add two numbers $a$ and $b$, the result is called the sum of $a$ and $b$. The numbers $a$ and $b$ are the terms of the sum.
Commutative Law for Addition: If $a$ and $b$ are numbers, then

Examples

• The sum "a number n plus 5" can be written as $n + 5$ or $5 + n$ because addition is commutative.
• The product "a number y times 3" is written as $3y$. This is the same as $y \cdot 3$, but putting the number first is standard.
• "The total of your score s and a bonus of 10 points" is $s + 10$, which is equivalent to $10 + s$.

Explanation

This is the "order doesn't matter" rule. For addition and multiplication, you can swap the numbers' positions and still get the same answer. This handy property is called the Commutative Law, and it's a great tool for rearranging expressions.

Property

When we subtract $b$ from $a$, the result is called the difference of $a$ and $b$. As with addition, a and b are called terms. The operation of subtraction is not commutative.

When we divide $a$ by $b$, the result is called the quotient of $a$ and $b$. We call a the dividend and b the divisor. In algebra we indicate division by using the division symbol, $\div$, or a fraction bar. The operation of division is not commutative.

Examples

• The phrase "7 subtracted from a number $x$" must be written as $x - 7$. The expression $7 - x$ represents "$x$ subtracted from 7".
• "A number $p$ divided by 10" is written as the quotient $\frac{p}{10}$. The reverse, $\frac{10}{p}$, means "10 divided by $p$".
• "Your height $h$ reduced by 2 inches" is expressed as the difference $h - 2$.

Property

Substituting a specific value for a variable into an expression and calculating the result is called evaluating the expression. This process turns a general algebraic rule into a specific numerical answer.

Examples

• To evaluate the expression $4k - 1$ when $k = 3$, we substitute 3 for $k$: $4(3) - 1 = 12 - 1 = 11$.
• Find the value of $\frac{m}{5} + 2$ for $m = 20$. We calculate $\frac{20}{5} + 2 = 4 + 2 = 6$.
• If a taxi fare is $2d + 3$ where d is distance in miles, a 5-mile trip costs $2(5) + 3 = 10 + 3 = 13$ dollars.

Explanation

Evaluating an expression is like using a recipe. The expression is the general recipe, and the given value for the variable is the specific ingredient. You just plug in the number and do the math to find the final result.

Property

Distance: To find the distance traveled, multiply the rate by the time.

Examples

• A car travels at a rate ($r$) of 50 mph for a time ($t$) of 4 hours. The distance is $d = rt = 50(4) = 200$ miles.
• A bake sale had revenue ($R$) of 250 dollars and costs ($C$) of 50 dollars. The profit is $P = R - C = 250 - 50 = 200$ dollars.
• On three tests, you score 80, 90, and 100. The average is $A = \frac{S}{n} = \frac{80+90+100}{3} = \frac{270}{3} = 90$.

Explanation

Formulas are ready-to-use equations for solving common real-world problems. Just identify the right formula for your situation, plug in the known values, and solve for the unknown quantity. They are powerful shortcuts in math!