Expressions, Functions & Data · 8 min read
Expressions and Formulas
Algebra uses expressions and formulas to describe real situations. This chapter reviews the properties and procedures that every later topic relies on, then covers two skills the test asks about directly: explaining what each part of an expression means in context, and rearranging a formula to isolate a chosen variable.
Properties, Order of Operations and Simplifying
The properties of real numbers are the reasons behind each step when an expression is rewritten. Subtraction and division are neither commutative nor associative, so order and grouping matter for those operations.
| Property | Rule | Example |
|---|---|---|
| Commutative | ~a + b = b + a~ and ~ab = ba~ | ~3 + 7 = 7 + 3~ |
| Associative | ~(a + b) + c = a + (b + c)~ and ~(ab)c = a(bc)~ | ~(2 · 3) · 4 = 2 · (3 · 4)~ |
| Distributive | ~a(b + c) = ab + ac~ | ~5(8 − 2) = 40 − 10 = 30~ |
| Identity | ~a + 0 = a~ and ~a · 1 = a~ | ~15 + 0 = 15~ |
| Inverse | ~a + (−a) = 0~ and ~a · [[1//a]] = 1~, where ~a ≠ 0~ | ~5 · [[1//5]] = 1~ |
| Zero property | ~a · 0 = 0~ | ~−9 · 0 = 0~ |
When an expression has several operations, follow the order of operations: first grouping symbols, working from the innermost set outward and treating a fraction bar or absolute value bars as grouping symbols; then exponents; then multiplication and division from left to right; then addition and subtraction from left to right. Multiplication does not come before division: ~24 ÷ 6 · 2 = 4 · 2 = 8~. To evaluate an expression, replace each variable with its value in parentheses and then follow that order.
- Parentheses, then the exponent: ~3 + 4 · 4^{2} ÷ 8 = 3 + 4 · 16 ÷ 8~.
- Multiply and divide from left to right: ~4 · 16 = 64~ and ~64 ÷ 8 = 8~. Then add: ~3 + 8 = 11~.
- Substitute ~−3~ in parentheses: ~(−3)^{2} − 2(−3) + 1 = 9 + 6 + 1 = 16~.
- Answer: 11 and 16
To simplify an expression, distribute to remove parentheses and then combine like terms. Like terms have the same variables raised to the same powers: ~3x~ and ~7x~ are like terms, but ~3x~ and ~3x^{2}~ are not. Combine like terms by adding or subtracting their coefficients, as in ~3a + 2b − 5a + 4b + a = −a + 6b~.
- Distribute 4: ~8x + 12~.
- Distribute ~−2~, keeping its sign: ~−2x + 10~.
- Combine like terms: ~8x − 2x = 6x~ and ~12 + 10 = 22~.
- Answer: ~6x + 22~
Interpreting Parts of an Expression
On the test you may be given an expression or equation that models a situation and asked what one piece of it represents. Start by naming the parts. The examples in the table refer to ~C = 45 + 0.25m~, the cost in dollars of renting a truck and driving it ~m~ miles.
| Part | What it is | In ~45 + 0.25m~ |
|---|---|---|
| Term | a part separated from the others by + or − | 45 and ~0.25m~ |
| Constant | a term with no variable; usually a starting or fixed amount | 45: a flat fee of $45 |
| Coefficient | the number multiplying a variable; usually a rate | 0.25: $0.25 per mile |
| Factor | a quantity that is multiplied by another | 0.25 and ~m~ |
To decide what a number means, ask what happens when the variable is 0 (this reveals the constant) and what happens when the variable increases by 1 (this reveals the coefficient). Checking units also helps: dollars = dollars + (dollars per mile)(miles). A group in parentheses, such as ~(n − 2)~ or ~(1 + r)~, should be read as a single quantity before you look at what multiplies it.
- Treat ~(n − 2)~ as one quantity: the total number of hours minus the 2 hours already covered. It is the number of additional hours.
- The factor 8 is the rate for each additional hour, so ~8(n − 2)~ is the total charge for the additional hours.
- Check with ~n = 5~: ~15 + 8(3) = 39~ dollars, which is $15 plus 3 additional hours at $8.
- Answer: ~(n − 2)~ is the number of hours beyond the first two; ~8(n − 2)~ is the cost of those hours.
- When ~t = 0~, ~(1.04)^{0} = 1~, so the value is 1,500. This is the starting balance.
- View the base as one quantity: ~1.04 = 1 + 0.04~. Each year the balance is multiplied by 1.04, which keeps 100% of the balance and adds 4% more.
- Answer: $1,500 is the initial deposit. 1.04 is the yearly growth factor; the balance grows 4% per year.
Factored expressions carry meaning too. If a rectangular garden has a width of ~w~ feet and an area of ~w(w + 5)~ square feet, the factor ~(w + 5)~ is the length, and it shows that the length is 5 feet more than the width. In ~0.80(p − 5)~, the group ~(p − 5)~ is a price after a $5 coupon, and the factor 0.80 means the customer then pays 80% of that amount, a 20% discount.
Rearranging Formulas
A formula is an equation with more than one variable. Rearranging a formula means isolating one chosen variable while treating every other letter as if it were a number. Use the same inverse operations that solve a one-variable equation, and apply each one to both sides: move every term that does not contain the chosen variable to the other side, factor the variable out if it appears in more than one term, then divide by whatever multiplies it. Undo a square with a square root.
- For ~t~: subtract ~P~ from both sides to get ~A − P = P~~rt~, then divide both sides by ~Pr~: ~t = [[A − P//Pr]]~.
- For ~P~: the variable appears in two terms, so factor it out: ~A = P(1 + rt)~.
- Divide both sides by the group ~(1 + rt)~: ~P = [[A//1 + rt]]~.
- Check with ~P = 1,000~, ~r = 0.05~, ~t = 4~ and ~A = 1,200~: ~[[200//50]] = 4~ and ~[[1,200//1.2]] = 1,000~.
- Answer: ~t = [[A − P//Pr]]~ and ~P = [[A//1 + rt]]~
- Subtract 32 from both sides: ~F − 32 = [[9//5]]C~.
- Multiply both sides by the reciprocal ~[[5//9]]~: ~[[5//9]](F − 32) = C~.
- Check: ~F = 212~ gives ~C = [[5//9]](180) = 100~.
- Answer: ~C = [[5//9]](F − 32)~
- Slope-intercept form is ~y = mx + b~, so solve for ~y~. Subtract ~3x~ from both sides: ~4y = −3x + 24~.
- Divide every term by 4: ~y = −[[3//4]]x + 6~.
- In general, when ~B ≠ 0~, ~Ax + By = C~ becomes ~y = −[[A//B]]x + [[C//B]]~.
- Answer: ~y = −[[3//4]]x + 6~, with slope ~−[[3//4]]~ and y-intercept 6
- Divide both sides of ~C = 2πr~ by ~2π~: ~r = [[C//2π]]~.
- Divide both sides of ~A = πr^{2}~ by ~π~: ~r^{2} = [[A//π]]~.
- Take the square root of both sides. A radius is positive, so keep only the positive root: ~r = √{A ÷ π}~.
- Answer: ~r = [[C//2π]]~ and ~r = √{A ÷ π}~
- Order of operations: grouping symbols, exponents, multiplication and division left to right, addition and subtraction left to right. Substitute negative values in parentheses.
- A constant is usually a starting or fixed amount; a coefficient is usually a rate per unit.
- Read a group such as ~(n − 2)~ or ~(1 + r)~ as one quantity. A factor of ~(1 + r)~ means growth at rate ~r~.
- To isolate a variable, undo operations in reverse order on both sides; factor first if the variable appears twice.
- After rearranging, check the result by substituting simple numbers.
Finished reading?
Lock it in with a quick Expressions, Functions & Data quiz.