Variables and expressions Knowledge-Based Study

Study notes

1. In arithmetic, we work with specific numbers such as 5, 12, or $\frac{1}{2}$. In algebra, we often need to represent unknown values, varying quantities, or general relationships.

   A variable is a symbol, typically a letter like $x$, $y$, or $n$, that represents a number whose value can vary or is not yet known.

   An algebraic expression is a mathematical phrase that combines numbers, variables, and operation symbols (addition, subtraction, multiplication, and division). Unlike an equation, an expression does not contain an equals sign ($=$).

   • Numerical expression: $8 + 3 \times 2$ (contains only numbers and operations)
   • Algebraic expression: $3x - 7$ (contains at least one variable)

   Understanding the structure of an expression helps in reading and simplifying it accurately.

   • Term: A single number, a variable, or numbers and variables multiplied together, separated by addition or subtraction signs. For example, in the expression $5x + 3y - 9$, the terms are $5x$, $3y$, and $-9$.
   • Coefficient: The numerical factor that multiplies a variable. In the term $7n$, the coefficient is 7. If a variable appears alone, such as $x$, its coefficient is implicitly 1 because $x = 1 \cdot x$.
   • Constant: A term that has a fixed numerical value and contains no variables. In $4a + 11$, the constant term is 11.

   Mathematical relationships are frequently written in everyday language. Translating these statements into algebraic expressions requires identifying the operation indicated by key words.

   • Addition: sum, plus, increased by, more than (e.g., "6 more than $k$" becomes $k + 6$)
   • Subtraction: difference, minus, decreased by, less than (e.g., "4 less than $p$" becomes $p - 4$)
   • Multiplication: product, times, multiplied by, twice (e.g., "the product of 8 and $m$" is written as $8m$)
   • Division: quotient, divided by, ratio of (e.g., "the quotient of $y$ and 5" is written as $\frac{y}{5}$)

   Caution: Be careful with subtraction phrases like "less than." The phrase "9 less than $x$" translates to $x - 9$, not $9 - x$.

   To evaluate an algebraic expression means to find its numerical value when each variable is replaced with a specific number.

   Follow two steps:

   1. Substitute: Replace each variable with its given value, using parentheses where appropriate to preserve correct multiplication or signs.
   2. Calculate: Compute the resulting numerical expression following the standard order of operations (parentheses, exponents, multiplication and division from left to right, addition and subtraction from left to right).

   Example

   Evaluate the expression $4a - 3b + 5$ when $a = 6$ and $b = 2$:

   Step 1 (Substitute): $4(6) - 3(2) + 5$
   Step 2 (Multiply): $24 - 6 + 5$
   Step 3 (Add and subtract left to right): $18 + 5 = 23$