Lesson 2: Evaluating Expressions and Combining Like Terms

New Concept

When you replace the variables in an expression with selected numbers and simplify using the order of operations, you have evaluated the expression.

Why it matters

Algebra provides the language to describe complex systems, from physics to finance. Evaluating expressions is the critical skill that translates these abstract models into concrete, verifiable predictions.

What’s next

Next, you'll practice substituting values for variables and combining like terms to simplify expressions.

When you replace the variables in an expression with selected numbers and simplify using the order of operations, you have evaluated the expression.

Evaluate $a^2b - b$ if $a = -3$ and $b = -5$. Solution: $(-3)^2(-5) - (-5) = (9)(-5) - (-5) = -45 + 5 = -40$.
Evaluate $3ab + 2a^2$ if $a = -2$ and $b = 5$. Solution: $3(-2)(5) + 2(-2)^2 = -30 + 2(4) = -30 + 8 = -22$.

Think of it like a secret recipe! An expression is the recipe with mystery ingredients 'x' and 'y'. To 'evaluate' it, you swap those ingredients with actual numbers and follow the order of operations to see what delicious numerical result you cook up. It’s all about finding that one final value hiding in the math.

Like terms have the same variable raised to the same power. To simplify, add the coefficients of the like terms.

Simplify: $4ab - 3a + 5 - 7ba + 6a$. Solution: $(4ab - 7ab) + (-3a + 6a) + 5 = -3ab + 3a + 5$.
Simplify: $8p^2 + 2q - 5p^2 - 3q + 10$. Solution: $(8p^2 - 5p^2) + (2q - 3q) + 10 = 3p^2 - q + 10$.

Imagine you have a big pile of fruit. You can easily count how many apples ($x^2$) you have and how many bananas ($xy$) you have, but you cannot add them into one group of 'apple-bananas'. Like terms are the same type of fruit. You can only combine things that are exactly alike to tidy up your expression!

When using your calculator to square negative numbers you need to use parentheses. For example, $(-3)^2 = 9$, but $-3^2 = -9$.

For $x=-4$, $x^2$ is evaluated as $(-4)^2 = (-4) \cdot (-4) = 16$.
Be careful not to write $-4^2$, which means $-(4 \cdot 4) = -16$.
Evaluate $k^2 - k$ for $k = -6$. Solution: $(-6)^2 - (-6) = 36 + 6 = 42$.

Parentheses act like a protective force field for negative signs. When you write $(-5)^2$, you're telling the math world to square the entire thing, negative sign and all, resulting in a positive 25. But without that force field, $-5^2$ means you only square the 5, and the negative sign just waits to get tacked on at the end.

1. Parentheses and grouping symbols 2. Exponents 3. Multiply and divide from left to right 4. Add and subtract from left to right.

Simplify $20 - 4 \cdot (7 - 5)^2 \div 8$. Solution: $20 - 4 \cdot (2)^2 \div 8 = 20 - 4 \cdot 4 \div 8 = 20 - 16 \div 8 = 20 - 2 = 18$.
Evaluate $x(y-z) + z^3$ for $x=5, y=4, z=2$. Solution: $5(4-2) + 2^3 = 5(2) + 8 = 10 + 8 = 18$.

Remember the powerful phrase 'Please Excuse My Dear Aunt Sally'! It's the ultimate guide for navigating complex math problems. First, handle anything in Parentheses. Next, conquer all the Exponents. Then, team up Multiplication and Division (working left to right). Finally, finish with Addition and Subtraction (also left to right). This order ensures everyone gets the same correct answer.