1.1 Use the Language of Algebra

Algebraic expressions and equations are the building blocks of algebra. They use variables, constants, and symbols to represent mathematical relationships and operations. Understanding how to read and manipulate these expressions is crucial for everything you'll do in this course.

This section covers factors, multiples, prime factorization, variables, order of operations, substitution, combining like terms, and translating word problems into algebra.

Algebraic Expressions and Equations

Factors and multiples of expressions

Factors are numbers or expressions that multiply together to produce a given number or expression. For example, the factors of 12 are 1, 2, 3, 4, 6, and 12 because each divides evenly into 12.

The greatest common factor (GCF) is the largest factor shared by two or more numbers. To find the GCF of 24 and 36, list the factors of each and pick the largest one they share: 12.

Prime factorization breaks a number down into a product of prime numbers (numbers whose only factors are 1 and themselves). To prime factorize a number:

1. Divide by the smallest prime (2, 3, 5, 7...) that goes in evenly
2. Keep dividing the result until every factor is prime
3. Write the original number as the product of all those primes

For example, $60 = 2 \times 2 \times 3 \times 5 = 2^2 \times 3 \times 5$.

The least common multiple (LCM) is the smallest number divisible by all the given numbers. To find it:

1. Find the prime factorization of each number
2. For each prime factor, take the highest power that appears in any factorization
3. Multiply those together

For 12 ($2^2 \times 3$) and 18 ($2 \times 3^2$), the LCM is $2^2 \times 3^2 = 36$.

Variables as representations

Variables ($x$, $y$, $z$) represent unknown or changing quantities. They can take on any value from a given set of numbers.

Constants are fixed values that don't change within an expression or equation. In $3x + 7$, the 7 is a constant and the 3 is the coefficient (the number multiplied by a variable).

Algebraic symbols represent operations ($+$, $-$, $\times$, $\div$) and relationships ($=$, $\neq$, $>$, $<$). Parentheses $( )$ group terms and indicate which operations to perform first.

Variables and symbols let you model real-world situations. For example, if a rectangle's length is 3 units more than twice its width $w$, the length is $2w + 3$. The perimeter becomes:

$2(2w + 3) + 2w = 4w + 6 + 2w = 6w + 6$

Order of operations in algebra

When an expression has multiple operations, you need a consistent rule for which to do first. That rule is PEMDAS:

1. Parentheses: simplify everything inside grouping symbols first
2. Exponents: evaluate powers and roots
3. Multiplication and Division: work left to right
4. Addition and Subtraction: work left to right

A common mistake is treating multiplication as always before division (or addition before subtraction). Multiplication and division have equal priority, so you just go left to right. Same for addition and subtraction.

For example, $8 \div 2 \times 4 = 4 \times 4 = 16$, not $8 \div 8 = 1$.

Substitution in expressions

Substitution means replacing variables with specific values and then simplifying. Follow order of operations carefully after substituting.

Evaluate $2x^2 - 3y + 4$ when $x = 2$ and $y = 1$:

1. Substitute: $2(2)^2 - 3(1) + 4$

2. Exponents first: $2(4) - 3(1) + 4$

3. Multiply: $8 - 3 + 4$

4. Add/subtract left to right: $5 + 4 = 9$

A common error with substitution: if $x = -3$, then $x^2 = (-3)^2 = 9$, but $-x^2 = -(3^2) = -9$. The placement of the negative sign matters.

Like terms in expressions

Like terms have the same variable(s) raised to the same power(s). For instance, $3x^2$ and $-4x^2$ are like terms, but $3x^2$ and $3x$ are not because the exponents differ.

To combine like terms, add or subtract their coefficients while keeping the variable part the same.

$3x^2 + 2x - 4x^2 + 5x$

• Combine the $x^2$ terms: $3x^2 + (-4x^2) = -x^2$
• Combine the $x$ terms: $2x + 5x = 7x$
• Result: $-x^2 + 7x$

Constants are like terms with each other too. So in $4x + 3 - 2x + 7$, you'd combine $4x$ and $-2x$ to get $2x$, and combine $3$ and $7$ to get $10$, giving you $2x + 10$.

Word problems to algebraic equations

Translating word problems into algebra is one of the most practical skills you'll build. Here's the process:

1. Identify the unknown and assign it a variable

2. Translate keywords into operations:

   • "Sum," "more than," "increased by" $\rightarrow$ addition
   • "Difference," "less than," "decreased by" $\rightarrow$ subtraction
   • "Product," "times," "of" $\rightarrow$ multiplication
   • "Quotient," "divided by," "per" $\rightarrow$ division
   • "Equals," "is," "was," "will be" $\rightarrow$ $=$

3. Write the equation and solve

Watch out for "less than" because it reverses the order. "Five less than a number" is $x - 5$, not $5 - x$.

Example: The sum of three consecutive even integers is 84. Find the integers.

• Let $x$ = the first even integer. The next two are $x + 2$ and $x + 4$.
• Translate: $x + (x + 2) + (x + 4) = 84$
• Simplify: $3x + 6 = 84$
• Solve: $3x = 78$, so $x = 26$
• The three consecutive even integers are 26, 28, and 30.

Additional Algebraic Concepts

Algebraic properties govern how you can rearrange and simplify expressions:

• Commutative property: order doesn't matter for addition or multiplication ($a + b = b + a$)
• Associative property: grouping doesn't matter for addition or multiplication ($(a + b) + c = a + (b + c)$)
• Distributive property: multiplication distributes over addition ($a(b + c) = ab + ac$)

Inequalities use symbols like $>$, $<$, $\geq$, and $\leq$ to show that one quantity is greater or less than another, rather than equal.

Function notation $f(x)$ represents a relationship where each input $x$ produces exactly one output. If $f(x) = 2x + 1$, then $f(3) = 2(3) + 1 = 7$. You'll work with this notation much more in later units.