Lesson 1: Working with Variables

New Concept

We simplify algebraic expressions by combining "like terms"—terms with identical variable parts. To combine them, simply add or subtract their coefficients. This is the key to making complex expressions manageable.

What’s next

Next, you'll see this in action with worked examples and interactive practice cards to master combining like terms and evaluating expressions.

Property

Like terms are terms where the variable part is the same. The numbers multiplied by the variable are called the coefficients.

To add or subtract like terms:

1. Add or subtract the coefficients.
2. Do not change the variable part of the terms.

Examples

• To combine $8m - 3m$, we subtract the coefficients: $(8-3)m = 5m$.
• To combine $7ab + 5ab$, we add the coefficients: $(7+5)ab = 12ab$.
• To combine $y + 9y$, remember the coefficient of $y$ is 1. So, $(1+9)y = 10y$.

Property

Equivalent expressions have the same value when we substitute a number for the variable. We cannot combine unlike terms. If the variable parts of the terms are not identical, they are not like terms, and they cannot be combined.

Examples

• The expressions $6x + 2x$ and $8x$ are equivalent. If $x=3$, both expressions equal 24.
• The expressions $6 + 2x$ and $8x$ are not equivalent. If $x=3$, $6+2x$ is 12, while $8x$ is 24.
• In the expression $9a + 4b - 3a$, we can combine the like terms to get the equivalent expression $6a + 4b$.

Explanation

Equivalent expressions are different ways to write the same mathematical idea. For example, $2+3$ and $5$ are equivalent. Simplifying an expression means finding a shorter, equivalent version, which makes it easier to work with.

Property

To multiply a term by a constant, we multiply the coefficient of the term by the constant.

Examples

• To simplify $7(3y)$, multiply the constant and the coefficient: $7 \cdot 3y = 21y$.
• A choir has 15 members, and each must sell $t$ tickets. The total tickets to be sold is $15t$. If each ticket costs 8 dollars, the total revenue is $8(15t) = 120t$ dollars.
• To simplify $0.25(12b)$, multiply the numbers: $0.25 \cdot 12b = 3b$.

Explanation

Multiplying a term by a constant is like scaling up a group. If one kit contains $4p$ parts, then 5 kits will contain $5 \times 4p = 20p$ parts. You simply multiply the numbers at the front of the term.

Property

In like terms, the variable parts of the terms must be exactly the same, including their exponents. When combining like terms, we do not add the exponents.

Examples

• To combine $5x^3 + 2x^3$, we add the coefficients because the variable parts are identical: $(5+2)x^3 = 7x^3$.
• The expression $8a^2 + 3a$ cannot be simplified because $a^2$ and $a$ are not like terms.
• Be careful not to add exponents. $5x^3 + 2x^3$ is $7x^3$, not $7x^6$.

Explanation

For terms to be 'like,' their variable parts must be identical twins, including the powers. You can add $x^2$ to another $x^2$, but you cannot add $x^2$ to $x^3$ because they represent different quantities, like a square and a cube.

Property

We can rearrange factors in any order and get the same product. To multiply variable expressions, rearrange the factors to put the coefficients together and the variable factors together.

Examples

• To multiply $(5x)(6x)$, rearrange the factors: $(5 \cdot 6)(x \cdot x) = 30x^2$.
• To multiply $(4a)(2b)$, rearrange the factors: $(4 \cdot 2)(a \cdot b) = 8ab$.
• To multiply $(3y^2)(7y)$, rearrange the factors: $(3 \cdot 7)(y^2 \cdot y) = 21y^3$.

Explanation

When multiplying terms, you can shuffle all the pieces. Group the numbers and multiply them. Then, group the variables and multiply them. Multiplying identical variables means adding their exponents, like $a \cdot a = a^2$.

Property

When we evaluate an algebraic expression, we follow the order of operations.
Order of Operations:

1. Perform any operations inside parentheses, or above or below a fraction bar.
2. Compute all powers and roots.
3. Perform all multiplications and divisions from left to right.
4. Perform additions and subtractions from left to right.

Examples

• Evaluate $30 - 4x^2$ for $x=2$. Substitute: $30 - 4(2)^2$. Power first: $30 - 4(4)$. Then multiply: $30 - 16$. Finally, subtract: $14$.
• Evaluate $(10-c)^2$ for $c=3$. Substitute: $(10-3)^2$. Parentheses first: $(7)^2$. Then compute the power: $49$.
• Evaluate $2(a^2 + 1)$ for $a=5$. Substitute: $2(5^2+1)$. Inside parentheses, do the power first: $2(25+1)$. Then add: $2(26)$. Finally, multiply: $52$.

Explanation

Evaluating an expression means finding its final numerical value. The Order of Operations (PEMDAS) is the universal rulebook that ensures everyone gets the same answer from the same calculation. Always follow the steps precisely.