Expressions, Substitution and Mathematical Language, SAT math lesson

A gym charges a one-time fee of 25 dollars and then 30 dollars for each month. If $m$ is the number of months, the total cost in dollars is $25 + 30m$ . That short string holds a whole sentence. The 25 is the fee you pay once, the 30 is the price of one month, and $m$ is the number you choose. Many SAT questions ask exactly this kind of thing, what a number in an expression means, so it helps to read algebra the way you read words.

See also Reading Formulas and Substituting Values in Foundations for Calculus and Function Notation and Evaluation in Precalculus.

Key terms

Variable

Definition A variable is a letter or other symbol that stands for a number from a set of allowed values. On the SAT, variables stand for real numbers unless a question says otherwise. When a variable comes from a situation, it needs a stated meaning and a unit, for example ” $m$ is the number of months” or ” $t$ is the time in seconds after the ball is thrown”. Without that, a value of the variable has no meaning.

In plain words A variable is a labeled box for a number. The label has to say what is being counted or measured and in what unit.

Example Let $h$ be the height of a plant in centimeters and $w$ the number of weeks since it was planted, and suppose $h = 4 + 1.5w$ . After 6 weeks

h = 4 + 1.5 \cdot 6 = 13,

so the plant is 13 centimeters tall. The same number 13 would mean nothing if nobody had said that $h$ is a height in centimeters.

Constant

Definition A constant is a quantity whose value stays fixed throughout a problem. It can be a specific number such as $7$ or $\pi$ , or a letter that names a fixed but possibly unknown number, as in ” $y = kx$ , where $k$ is a constant”. A letter used as a constant has the same value for every value of the variables.

In plain words Variables change from one case to the next. Constants do not, even when they are written as letters.

Example Suppose $y = kx$ for a constant $k$ , and the point $(2, 10)$ fits the equation. Then $10 = 2k$ , so $k = 5$ . That value is now fixed, so at $x = 7$ the equation gives $y = 5 \cdot 7 = 35$ . In the gym cost $25 + 30m$ the number $25$ is a constant, the same for every number of months.

Expression

Definition An algebraic expression is a combination of numbers, variables and operations (addition, subtraction, multiplication, division, powers and roots), possibly with parentheses, that has a numerical value once each variable is given a value. An expression contains no equals sign and no inequality symbol. Two expressions are equivalent when they have the same value for every allowed value of their variables.

To show that two expressions are not equivalent, one value where they differ is enough. To show that they are equivalent, checking a few values is not enough, and you need algebra that rewrites one as the other.

In plain words An expression is a phrase, not a sentence. It names a quantity but claims nothing, so you can evaluate it or simplify it, but there is nothing to solve.

Example Compare $(x + 3)^2$ with $x^2 + 9$ . At $x = 0$ both equal $9$ , which proves nothing. At $x = 1$ the first is $16$ and the second is $10$ , so they are not equivalent. Now compare $2(x + 3) - 4$ with $2x + 2$ . Algebra settles it,

\begin{aligned} 2(x + 3) - 4 &= 2x + 6 - 4 \\[0.4em] &= 2x + 2, \end{aligned}

so these two are equivalent for every real $x$ .

Term

Definition When an expression is written as a sum, its terms are the parts being added. Subtraction counts as adding the opposite, so the sign in front of a term belongs to that term. In $4x^2 - 7x + 2$ the terms are $4x^2$ , $-7x$ and $2$ . A term with no variable is a constant term. Like terms have the same variables raised to the same powers, and only like terms can be combined into a single term.

In plain words Terms are the pieces between the plus and minus signs, each carrying its own sign. A product in parentheses, such as $3(x + 2)$ , counts as one term until you expand it.

Example In $5x - 3y - 2x + 8$ the terms are $5x$ , $-3y$ , $-2x$ and $8$ . The like terms $5x$ and $-2x$ combine to $3x$ , so the expression simplifies to $3x - 3y + 8$ . The terms $3x$ and $-3y$ cannot be combined, since $x$ and $y$ are different variables.

Coefficient

Definition In a term that is a product of a number and one or more variables, the numerical factor is the coefficient of that term. The coefficient of $-7x$ is $-7$ , the coefficient of $x^2$ is $1$ , and the coefficient of $-x$ is $-1$ . A constant letter can also play this role, as $a$ does in $ax + b$ . In a linear expression such as $ax + b$ , the coefficient $a$ is the amount the expression changes when $x$ increases by $1$ , so in a context it is a rate.

In plain words The coefficient is the number glued to the variable. In a real situation it is usually a rate, something per unit.

Example A phone plan costs $12 + 0.05t$ dollars in a month with $t$ text messages. The coefficient $0.05$ is the cost in dollars of each text, because each extra text adds $0.05$ to the total. The constant term $12$ is the fixed monthly charge, the cost when $t = 0$ . With 80 texts the bill is

12 + 0.05 \cdot 80 = 16 \text{ dollars.}

Factor

Definition The factors of a product are the quantities that are multiplied to form it. In the term $6xy$ the numbers and variables $6$ , $x$ and $y$ are factors, and so are products of them such as $3x$ . In $3(x + 2)$ the factors are $3$ and $x + 2$ . To factor an expression means to rewrite a sum as an equivalent product, as in $6x + 9 = 3(2x + 3)$ .

In plain words Terms are added and factors are multiplied. Whether something is a term or a factor depends on the operation that joins it to the rest.

Example In $2(x - 5) + 3x$ there are two terms, $2(x - 5)$ and $3x$ . The quantity $x - 5$ is a factor of the first term but not of the whole expression. After expanding,

\begin{aligned} 2(x - 5) + 3x &= 5x - 10 \\[0.4em] &= 5(x - 2), \end{aligned}

and now the whole expression is one product with factors $5$ and $x - 2$ .

Equation

Definition An equation is a statement that two expressions are equal, written with the symbol $=$ . An equation in a variable can be true for some values of the variable and false for others. A value that makes it true is a solution, and solving the equation means finding all of its solutions. An equation that is true for every allowed value of its variables is an identity.

An expression is not an equation. An expression has a value, while an equation is a claim that is either true or false for each value of the variable.

In plain words An equation is a full sentence with a verb, the equals sign. You can ask whether it is true, and for which numbers.

Example The expression $2x + 6$ has the value $14$ at $x = 4$ , and there is nothing to solve. The equation $2x + 6 = 14$ is true at $x = 4$ , since $8 + 6 = 14$ , and false at $x = 0$ , since $6 \ne 14$ . The equation $3(x + 2) - x = 2x + 6$ is an identity, because the left side simplifies to the right side for every real $x$ .

Substitution

Definition Substitution means replacing every occurrence of a variable in an expression or equation by a given number or expression, and then evaluating with the usual order of operations. The same value replaces the variable in every place it appears. Write the substituted value in parentheses, especially when it is negative or is itself an expression, so that powers and products apply to the whole value. Without them, $-2^2$ means $-(2^2) = -4$ , while $(-2)^2 = 4$ .

In plain words Take the variable out and put the number in its place, inside parentheses, everywhere at once. Then do the arithmetic.

Example Evaluate $3x^2 - 5x + 1$ at $x = -2$ . Write the original expression on the first line, and directly under it put $(-2)$ in each spot where $x$ was,

\begin{aligned} &3x^2 - 5x + 1 \\[0.4em] &3(-2)^2 - 5(-2) + 1 \\[0.4em] &\quad = 3 \cdot 4 + 10 + 1 = 23. \end{aligned}

Two slips are common here. Writing $3 \cdot -2^2$ gives $-12$ for the first term instead of $12$ , and writing $-5 - 2$ for the middle term gives $-7$ instead of $+10$ .

You can also substitute an expression. If $y = 2x - 1$ , then

3y + 4 = 3(2x - 1) + 4 = 6x + 1.

Check yourself

A delivery service charges $C = 4.50 + 1.25d$ dollars to deliver a package $d$ miles. What does $1.25$ represent in this equation? (A) The total charge for a delivery of 1 mile (B) The fixed charge for every delivery (C) The additional charge for each mile of the delivery (D) The number of miles that can be delivered for 4.50 dollars

Show the answer

The correct choice is (C). The number $1.25$ is the coefficient of $d$ , so each extra mile adds $1.25$ dollars to the charge. Choice (A) is tempting, but a 1 mile delivery costs $4.50 + 1.25 = 5.75$ dollars. The fixed charge in (B) is the constant term $4.50$ , and (D) mixes up dollars with miles.

If $x = -3$ , what is the value of $2x^2 - x - 7$ ?

Show the answer

The answer is $14$ . With parentheses around $-3$ ,

\begin{aligned} 2(-3)^2 - (-3) - 7 &= 18 + 3 - 7 \\[0.4em] &= 14. \end{aligned}

Squaring without parentheses gives $-18$ for the first term and a total of $-22$ , and dropping the sign on the middle term gives $18 - 3 - 7 = 8$ .

Which expression is equivalent to $4(x - 2) - (x - 5)$ ? (A) $3x - 3$ (B) $3x - 13$ (C) $3x + 3$ (D) $5x - 3$

Show the answer

The correct choice is (A). Distribute both the $4$ and the minus sign,

4x - 8 - x + 5 = 3x - 3.

Choice (B) comes from writing $-(x - 5)$ as $-x - 5$ , (C) from multiplying only the $x$ by $4$ , and (D) from writing $-(x - 5)$ as $+x + 5$ . Testing values shows the danger of checking only one. At $x = 0$ the original and (D) both equal $-3$ , yet at $x = 1$ the original is $0$ while (D) is $2$ . One value where two expressions differ proves they are not equivalent, but one value where they agree proves nothing.

Expressions, Substitution and Mathematical Language

Theory, part 9

Practice, part 10

Key terms

Variable

Constant

Expression

Term

Coefficient

Factor

Equation

Substitution

Check yourself

How this lesson connects

Builds on

Used later in