Lesson 9: Evaluating and Comparing Algebraic Expressions

New Concept

An algebraic expression is an expression with variables and/or numbers that uses operations (e.g., +, -, $\times$, or $\div$).

What’s next

This card is just the foundation! Soon, we’ll dive into worked examples on evaluating expressions and use them to solve practical problems.

Property

To evaluate an algebraic expression, substitute a value for each variable and simplify using the order of operations.

Examples

Evaluate $2x - 5x + ax$ for $x = 4$ and $a = 2$. $2(4) - 5(4) + (2)(4) = 8 - 20 + 8 = -4$.
Evaluate $5y + 2z - yz$ for $y = 3$ and $z = 5$. $5(3) + 2(5) - (3)(5) = 15 + 10 - 15 = 10$.

Explanation

Think of it like a recipe! You replace ingredient names (variables) with their actual amounts (numbers). Then, you follow the cooking steps (order of operations) to get your final, delicious dish. The result is the value of the expression for those specific numbers.

Property

To evaluate expressions with exponents, substitute the given values, then simplify using the order of operations (PEMDAS), paying close attention to parentheses and powers.

Examples

Evaluate $2(z - y)^2 - 3y^2$ for $z = 5, y = 3$. $2(5 - 3)^2 - 3(3)^2 = 2(2)^2 - 3(9) = 8 - 27 = -19$.
Evaluate $a^3 + (b - a)^2$ for $a = 2, b = 7$. $(2)^3 + (7 - 2)^2 = 8 + (5)^2 = 8 + 25 = 33$.

Explanation

When exponents join the party, they get VIP treatment! Always solve the powers right after substituting your numbers, especially those inside parentheses. This ensures everything else in the equation gets calculated in the correct, orderly fashion, preventing mathematical chaos and leading to the right answer.

Property

To compare two algebraic expressions, evaluate each one separately by substituting the given values for the variables. Then, use $<, >,$ or $=$ to compare the final results.

Examples

Compare 2x² + 3y with 5x + y² for x = 3, y = 4.30 < 31.

Compare a³ - b with 10ab for a = 4, b = 2.62 < 80.