Lesson 2: Expressions and Equations

New Concept

This lesson introduces algebraic expressions, showing how to translate real-world scenarios into mathematical language. You'll learn to write, simplify, and evaluate expressions involving variables and signed numbers, using the order of operations as your guide.

What’s next

Get ready to master these skills! Next, you'll tackle interactive examples on writing expressions and then solve practice problems evaluating them with signed numbers.

Property

When we write an algebraic expression, we use the same operations on a variable that we would use to calculate with a specific number. Writing down the expression for a specific numerical value can help us write an algebraic expression.

Examples

• A repair service charges 50 dollars for a visit plus 30 dollars per hour. If the job takes $h$ hours, the total cost is $50 + 30h$.

• A sweater is on sale for $25\%$ off its original price $p$. The sale price is the original price minus the discount, which is $p - 0.25p$, or $0.75p$.

Property

When we evaluate an algebraic expression, we substitute the given value for the variable and then we must follow the order of operations to simplify the resulting numerical expression. Parentheses indicate that we should perform the operations inside them first.

Examples

• To evaluate $5(x - 3)$ for $x = 9$, we substitute 9 for $x$ and simplify inside the parentheses first: $5(9 - 3) = 5(6) = 30$.

• A gym membership costs $25 + 10m$ dollars for $m$ months. For 6 months, the cost is $25 + 10(6) = 25 + 60 = 85$ dollars.

Property

The order of operations applies to signed numbers. Operations inside parentheses or other grouping devices are performed first. To simplify expressions, it can be helpful to rewrite subtractions as equivalent additions.

Examples

• To simplify $10 - 2 - [-4 + (-7) - (-3)]$, first work inside the brackets: $10 - 2 - [-4 - 7 + 3] = 10 - 2 - [-8] = 10 - 2 + 8 = 16$.

• Be careful to distinguish products from sums. For example, $4(-9)$ is a product, which equals $-36$. But $4 - 9$ is a sum, $4 + (-9)$, which equals $-5$.

Property

When we evaluate an algebraic expression at a negative number, we enclose the negative numbers in parentheses. This will help prevent us from confusing multiplication with subtraction.

Examples

• Evaluate $5x - 2xy$ for $x = -3$ and $y = 4$. Substitute with parentheses: $5(-3) - 2(-3)(4) = -15 - (-6)(4) = -15 - (-24) = -15 + 24 = 9$.

• Evaluate $a(b - 7)$ for $a = -5$ and $b = -2$. Substitute the values into the expression: $(-5)((-2) - 7) = (-5)(-9) = 45$.