Expressions and equations | ASVAB Algebra I

Now that you’ve reviewed the foundations of math, let’s move into algebra and the notation that comes with it.

Because algebra uses $x$ as a variable, multiplication is usually shown by writing terms side by side ( $3 x$ means $3 \times x$ ) or with a dot ( $x \cdot y$ ), rather than the $\times$ symbol - that way you don’t confuse a variable with a multiplication sign. Division is often written as a fraction or with $/$ .

Variables

Definitions

Variable: Non-number value in an equation (often a letter, like x or y)

Variables are symbols that stand in for numbers in a problem. Instead of writing a specific number, you use a variable when the value is unknown or can change.

Common variables include $x$ , $y$ , and $n$ .

For example, in the expression $x + 5$ , the value of $x$ is not known. If $x = 3$ , then the expression becomes $3 + 5 = 8$ .

Terms

Definitions

Term: An individual part of an equation or expression

Each part of an equation is a term. They can be numbers, variables, or a combination of the two.

The expression 3x² + 2x + 4 with each of its three terms, 3x², 2x and 4, boxed and labeled term. — Terms

A term can be:

A number
A variable
A number and a variable multiplied together

Like terms

Definitions

Like terms: Terms that have the same variable(s) raised to the same power
Polynomial: An expression made of one or more terms added or subtracted, where each variable is raised to a whole-number power (for example, $4 x^{3} + 2 x^{2} + 3$ )

Like terms are the only parts of a polynomial that can be combined. They must have identical variables raised to the same power. So $2 x^{3}$ can be combined with $3 x^{3}$ , but not $3 x^{2}$ because they have different powers. Combining like terms adds their coefficients and keeps the power: $2 x^{3} + 3 x^{3} = 5 x^{3}$ , not $5 x^{6}$ . Exponents are added only when powers of the same base are multiplied. Individual numbers can only be combined with other individual numbers.

Example:

$(4 x^{3} + 2 x^{2} + 3) + (7 x^{2} + 4)$

We can combine the $2 x^{2}$ and the $7 x^{2}$ because they have the same power. We can also combine $3$ and $4$ because they are both numbers. So this expression simplifies to:

$4 x^{3} + 9 x^{2} + 7$

Example:

$(10 x^{3} + 5 x) - (4 x^{3} + 2 x)$

Subtracting a polynomial means subtracting every term inside its parentheses, so distribute the negative sign across each term before combining:

$10 x^{3} + 5 x - 4 x^{3} - 2 x$

Now combine like terms ( $10 x^{3} - 4 x^{3}$ and $5 x - 2 x$ ):

$6 x^{3} + 3 x$

Knowledge check: Simplify $(17 p^{5} + 12 p - 6) - (12 + 5 p^{5} + 2 p^{2})$

(spoiler)

$12 p^{5} - 2 p^{2} + 12 p - 18$
(make sure to distribute the negative sign if you are having trouble getting this answer)

Coefficients

Definitions

Coefficient: The number in front of a variable. It tells you how many of that variable you have.

So $3 x$ means $x + x + x$ . Instead of having to write that out in a larger expression such as $x + x + x + y + y + 4$ , we can simplify it to say $3 x + 2 y + 4$ .

A coefficient is different from an exponent. An exponent is the small raised number attached to a variable, like $x^{3}$ , and tells you how many times the variable is multiplied by itself: $x^{3} = x \cdot x \cdot x$ .

Knowledge check: Which represents $4 y$ ? $(y + y + y + y)$ or $(y * y * y * y)$ ?

(spoiler)

$(y + y + y + y)$

One-step equations

One-step equations are exactly as advertised; they ask us to isolate a variable in just one step! This may mean that you have to multiply, divide, add, or subtract to get your variable by itself.

You just need to identify the operation the equation is using and “undo it” by performing the reverse operation on both sides.

Addition and subtraction are opposites of one another, and multiplication and division are opposites of one another. So, if the question has subtraction, you’ll need to use addition to find the answer.

Just remember that we’re always going to perform the same operation to both sides of the equals sign to keep the equation balanced!

Think: what is being done to the variable, and how can you undo it?

Example:

Solve the following for $x$ :

$x - 12 = 36$

Step 1: Find $x$ in the equation: $x - 12$ .

Step 2: Identify the operation being used: subtraction.

Step 3: Perform the opposite operation on both sides to undo the subtraction. Add $12$ to both sides.

Step 4: Simplify. $- 12 + 12 = 0$ , so $x$ is by itself on the left, and $36 + 12 = 48$ .

$x - 12 x - 12 (+ 12) x = 36 = 36 + 12 = 48$

Multiple variables

Sometimes equations will contain multiple, unique variables. This may look complicated, but don’t worry. You’ll typically either be given a value to plug in for one (or more) of your variables and asked to solve, or you’ll be asked to simply rearrange the equation to isolate one of your variables.

Sidenote

Substitution vs rearranging

Use substitution when you are given a value for a variable. Use rearranging (isolating) when you need to solve for one variable in terms of another.

Let’s look at an example of isolating one variable.

Example:

Solve for $y$ :

$2 x + 3 y = 6$

Step 1: Identify the variable to isolate: $y$ .

Step 2: Get $3 y$ by itself by subtracting $2 x$ from both sides:

$3 y = - 2 x + 6$

Step 3: Divide both sides by $3$ to isolate $y$ :

$\frac{3 y}{3} = \frac{- 2 x + 6}{3}$

Step 4: Simplify to get the final answer:

$y = \frac{- 2 x + 6}{3}$

If the number in front of $y$ is negative, dividing by it changes the sign of every term: $- 2 y = 4 x - 8$ gives $y = - 2 x + 4$ .

Here is the full solution shown together:

$2 x + 3 y 2 x + 3 y - 2 x 3 y \frac{3 y}{3} y = 6 = 6 - 2 x = - 2 x + 6 = \frac{- 2 x + 6}{3} = \frac{- 2 x + 6}{3}$

Since the ASVAB math sections don’t allow a calculator, leave an answer like this as a fraction - $\frac{- 2 x + 6}{3}$ - rather than trying to convert it to a decimal.

Substitution

Another common skill is substitution. In substitution problems, you’re given a value for a variable and you plug it into an expression.

Example:

If $x = 6$ , what is the value of $x^{2} - 17$ ?

$x^{2} - 17 6^{2} - 17 36 - 17 = 19$

Knowledge check: If $y = 4$ then $2 y + y = ?$

(spoiler)

$12$

Now that you’ve reviewed the foundations of math, let’s move into algebra and the notation that comes with it.

Variables

Definitions

Variable: Non-number value in an equation (often a letter, like x or y)

Variables are symbols that stand in for numbers in a problem. Instead of writing a specific number, you use a variable when the value is unknown or can change.

Common variables include $x$ , $y$ , and $n$ .

For example, in the expression $x + 5$ , the value of $x$ is not known. If $x = 3$ , then the expression becomes $3 + 5 = 8$ .

Terms

Definitions

Term: An individual part of an equation or expression

Each part of an equation is a term. They can be numbers, variables, or a combination of the two.

A term can be:

A number
A variable
A number and a variable multiplied together

Like terms

Definitions

Like terms: Terms that have the same variable(s) raised to the same power
Polynomial: An expression made of one or more terms added or subtracted, where each variable is raised to a whole-number power (for example, $4 x^{3} + 2 x^{2} + 3$ )

Example:

$(4 x^{3} + 2 x^{2} + 3) + (7 x^{2} + 4)$

We can combine the $2 x^{2}$ and the $7 x^{2}$ because they have the same power. We can also combine $3$ and $4$ because they are both numbers. So this expression simplifies to:

$4 x^{3} + 9 x^{2} + 7$

Example:

$(10 x^{3} + 5 x) - (4 x^{3} + 2 x)$

Subtracting a polynomial means subtracting every term inside its parentheses, so distribute the negative sign across each term before combining:

$10 x^{3} + 5 x - 4 x^{3} - 2 x$

Now combine like terms ( $10 x^{3} - 4 x^{3}$ and $5 x - 2 x$ ):

$6 x^{3} + 3 x$

Knowledge check: Simplify $(17 p^{5} + 12 p - 6) - (12 + 5 p^{5} + 2 p^{2})$

(spoiler)

$12 p^{5} - 2 p^{2} + 12 p - 18$
(make sure to distribute the negative sign if you are having trouble getting this answer)

Coefficients

Definitions

Coefficient: The number in front of a variable. It tells you how many of that variable you have.

So $3 x$ means $x + x + x$ . Instead of having to write that out in a larger expression such as $x + x + x + y + y + 4$ , we can simplify it to say $3 x + 2 y + 4$ .

Knowledge check: Which represents $4 y$ ? $(y + y + y + y)$ or $(y * y * y * y)$ ?

(spoiler)

$(y + y + y + y)$

One-step equations

One-step equations are exactly as advertised; they ask us to isolate a variable in just one step! This may mean that you have to multiply, divide, add, or subtract to get your variable by itself.

You just need to identify the operation the equation is using and “undo it” by performing the reverse operation on both sides.

Just remember that we’re always going to perform the same operation to both sides of the equals sign to keep the equation balanced!

Think: what is being done to the variable, and how can you undo it?

Example:

Solve the following for $x$ :

$x - 12 = 36$

Step 1: Find $x$ in the equation: $x - 12$ .

Step 2: Identify the operation being used: subtraction.

Step 3: Perform the opposite operation on both sides to undo the subtraction. Add $12$ to both sides.

Step 4: Simplify. $- 12 + 12 = 0$ , so $x$ is by itself on the left, and $36 + 12 = 48$ .

$x - 12 x - 12 (+ 12) x = 36 = 36 + 12 = 48$

Multiple variables

Sidenote

Substitution vs rearranging

Use substitution when you are given a value for a variable. Use rearranging (isolating) when you need to solve for one variable in terms of another.

Let’s look at an example of isolating one variable.

Example:

Solve for $y$ :

$2 x + 3 y = 6$

Step 1: Identify the variable to isolate: $y$ .

Step 2: Get $3 y$ by itself by subtracting $2 x$ from both sides:

$3 y = - 2 x + 6$

Step 3: Divide both sides by $3$ to isolate $y$ :

$\frac{3 y}{3} = \frac{- 2 x + 6}{3}$

Step 4: Simplify to get the final answer:

$y = \frac{- 2 x + 6}{3}$

If the number in front of $y$ is negative, dividing by it changes the sign of every term: $- 2 y = 4 x - 8$ gives $y = - 2 x + 4$ .

Here is the full solution shown together:

$2 x + 3 y 2 x + 3 y - 2 x 3 y \frac{3 y}{3} y = 6 = 6 - 2 x = - 2 x + 6 = \frac{- 2 x + 6}{3} = \frac{- 2 x + 6}{3}$

Since the ASVAB math sections don’t allow a calculator, leave an answer like this as a fraction - $\frac{- 2 x + 6}{3}$ - rather than trying to convert it to a decimal.

Substitution

Another common skill is substitution. In substitution problems, you’re given a value for a variable and you plug it into an expression.

Example:

If $x = 6$ , what is the value of $x^{2} - 17$ ?

$x^{2} - 17 6^{2} - 17 36 - 17 = 19$

Knowledge check: If $y = 4$ then $2 y + y = ?$

(spoiler)

$12$