Summary
Highlights
A variable is a symbol, typically a lowercase letter like x, y, or z, used to represent an unknown quantity or a placeholder. For example, if Mark earns $63 per day plus unknown tips (x), his total earnings can be represented as 63 + x.
If Mark's tips (x) are $7, his total earnings are $63 + $7 = $70. If his tips change to $9, his earnings become $63 + $9 = $72. Another example: if a gallon of milk costs an unknown amount (y) and Jason buys two gallons, the cost is 2 * y or 2y. If y is $1, he spends $2; if y is $3, he spends $6.
A 'term' is a single number, variable, or a number multiplied by one or more variables (e.g., 4x, 9y, 24xyz). A 'coefficient' is the numerical factor multiplying a variable (e.g., in 2y, 2 is the coefficient of y). A 'constant' is a number by itself whose value does not change (e.g., in 5x + 7, 7 is the constant).
An algebraic expression consists of one or more terms separated by plus or minus symbols (e.g., 5x + 2y - 4z). The value of an algebraic expression changes based on the values assigned to its variables. For instance, in 2x + 3, if x=4, the expression evaluates to 11 (2*4 + 3); if x=5, it evaluates to 13 (2*5 + 3); if x=-2, it evaluates to -1 (2*(-2) + 3).
The distributive property allows us to simplify expressions by multiplying a number outside parentheses by each term inside. For example, 3 * (x - 4) becomes 3x - 12. Similarly, 4 * (x - 7) + 2 becomes 4x - 28 + 2, which simplifies to 4x - 26.
Like terms are terms that have the exact same variable(s) raised to the exact same exponents. The coefficients can differ. For example, 5x and 3x are like terms, but 5x and 3y are not, nor are 2x and 3x³. When combining like terms, the variable part remains the same; only the coefficients are added or subtracted. For instance, 2x + 3x = 5x.
If terms are not alike (e.g., 2x + 3y), they cannot be combined. In an expression like 4x² + 3x² + z, only the like terms (4x² and 3x²) are combined, resulting in 7x² + z. A more complex example: -2(4x - 7) + 4(x - 3) first uses the distributive property to become -8x + 14 + 4x - 12. Then, combining like terms (-8x + 4x) and constants (14 - 12) yields -4x + 2.