Variables and Expressions

A rideshare app charges a $2.50 base fee plus $1.75 for every mile you travel. For a 3-mile trip that's $2.50 + $1.75(3) = $7.75. For an 8-mile trip, it's $2.50 + $1.75(8) = $16.50. Rather than recalculating from scratch every time, you can write one formula, cost = 2.50 + 1.75m, and plug in any number of miles, m. That formula is an algebraic expression, and m is a variable, the tool this whole lesson is built around.

What a variable is and why letters stand in for unknown or changing numbers. How to evaluate an expression by substituting a number for a variable. How to combine like terms to simplify expressions. How to translate a real-world situation into an algebraic expression.

A variable is a letter, most commonly x, n, or y, that represents a number we don't know yet or that can change. An expression is a combination of numbers, variables, and operations, like 4x + 7 or 3n - 2. Unlike an equation, an expression has no equals sign and no single fixed answer; its value depends entirely on what number you substitute for the variable. In the expression 4x + 7, the pieces separated by addition or subtraction are called terms: 4x is one term (a variable term with coefficient 4) and 7 is another term (a constant, since it never changes).

To evaluate an expression, replace the variable with a specific number and follow the order of operations (parentheses, exponents, multiplication/division, addition/subtraction, left to right). Example: evaluate 5x - 3 when x = 4. Step 1: Substitute: 5(4) - 3. Step 2: Multiply first: 20 - 3. Step 3: Subtract: 17. So when x = 4, the expression 5x - 3 equals 17. Change x to 10, and the same expression gives 5(10) - 3 = 47, a completely different value, because that's the whole point of a variable.

Like terms have the exact same variable raised to the exact same power, like 3x and 5x, or 2y^2 and -7y^2. You combine like terms by adding or subtracting their coefficients while keeping the variable part unchanged. Example: simplify 6x + 3 - 2x + 9. Group like terms: (6x - 2x) + (3 + 9). Combine: 4x + 12. Note that 4x and 12 are NOT like terms (one has a variable, one doesn't), so this expression is already fully simplified.

Algebra becomes powerful when you use it to model real life. A phone plan costing $30 per month plus $0.10 per text becomes 30 + 0.10t, where t is the number of texts. A rectangular garden that is 4 feet longer than it is wide has a width w and a length w + 4, so its perimeter is 2w + 2(w + 4), which simplifies to 4w + 8. Key phrases to watch for: "more than" or "increased by" usually means addition; "less than" means subtraction (and reverses order: "5 less than x" is x - 5, not 5 - x); "times" or "product of" means multiplication; "per" or "for each" signals a coefficient attached to a variable.