2.3.1 Expressions and equations

Achievable ASVAB

2. Mathematics Knowledge

2.3. Algebra I

Our ASVAB course is currently in development and is a work-in-progress.

Expressions and equations

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

Because algebra uses variables like $x$ and $y$ , it’s common to write multiplication with $*$ instead of $\times$ , and division with $/$ instead of $\div$ . This helps you avoid mixing up a variable (like $x$ ) with a multiplication symbol.

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

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. 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 equation becomes:

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

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)

Variables

Definitions

Polynomial: An expression with multiple terms that can’t be combined
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 $3 + 5 = 8$ .

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

Important to know!

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:

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

Here is the full solution shown together:

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

Sidenote

Don't worry

If this process doesn’t look familiar, keep reading. We’ll show you exactly how it’s done!

Coefficients

Definitions

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

What is a coefficient? A coefficient is the number in front of a variable. It represents the multiplier of that variable (in other words, it tells us how many of that particular variable we 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 $36 + 12 = 48$ .

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

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)

$12$

Algebra notation

Use $*$ for multiplication and $/$ for division instead of $\times$ / $\div$ to avoid confusion with variables like $x$

Terms & like terms

A term is a number, variable, or number-variable combo within an expression
Like terms share the same variable(s) raised to the same power (e.g., $2 x^{3}$ and $3 x^{3}$ combine; $3 x^{2}$ does not)
Only like terms can be combined in a polynomial; constants combine only with other constants
Remember to distribute negative signs when subtracting polynomials

Variables

Variables (e.g., $x$ , $y$ , $n$ ) represent unknown or changeable values
Substitution: plug in a given value for a variable and simplify
Isolating/rearranging: solve for one variable in terms of another (used when no value is given)
Isolating steps: move terms to isolate the variable’s term, then divide by its coefficient
- Example: $2 x + 3 y = 6$ solved for $y$ gives $y = \frac{- 2 x + 6}{3}$

Coefficients vs. exponents

Coefficient = number in front of a variable, shows how many of that variable (e.g., $3 x = x + x + x$ )
Exponent = small raised number showing repeated multiplication (e.g., $x^{3} = x \cdot x \cdot x$ )
Don’t confuse the two — they represent different operations

One-step equations

Solved by identifying the operation applied to the variable and undoing it with its inverse
Addition ↔ subtraction; multiplication ↔ division are inverse pairs
Always apply the same operation to both sides to keep the equation balanced

Substitution practice

Replace the variable with its given value, then simplify using order of operations
Example: if $x = 6$ , then $x^{2} - 17 = 36 - 17 = 19$

Next | 2.3.2 Rules and operations

Expressions and equations

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

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

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 equation becomes:

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

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)

Variables

Definitions

Polynomial: An expression with multiple terms that can’t be combined
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 $3 + 5 = 8$ .

Multiple variables

Sidenote

Important to know!

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:

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

Here is the full solution shown together:

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

Sidenote

Don't worry

If this process doesn’t look familiar, keep reading. We’ll show you exactly how it’s done!

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 $36 + 12 = 48$ .

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

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)

$12$

Achievable ASVAB

2. Mathematics Knowledge

2.3. Algebra I

Our ASVAB course is currently in development and is a work-in-progress.

Expressions and equations

6 min read

Font

Discuss

Feedback

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

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

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 equation becomes:

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

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)

Variables

Definitions

Polynomial: An expression with multiple terms that can’t be combined
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 $3 + 5 = 8$ .

Multiple variables

Sidenote

Important to know!

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:

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

Here is the full solution shown together:

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

Sidenote

Don't worry

If this process doesn’t look familiar, keep reading. We’ll show you exactly how it’s done!

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 $36 + 12 = 48$ .

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

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)

$12$

Algebra notation

Use $*$ for multiplication and $/$ for division instead of $\times$ / $\div$ to avoid confusion with variables like $x$

Terms & like terms

A term is a number, variable, or number-variable combo within an expression
Like terms share the same variable(s) raised to the same power (e.g., $2 x^{3}$ and $3 x^{3}$ combine; $3 x^{2}$ does not)
Only like terms can be combined in a polynomial; constants combine only with other constants
Remember to distribute negative signs when subtracting polynomials

Variables

Variables (e.g., $x$ , $y$ , $n$ ) represent unknown or changeable values
Substitution: plug in a given value for a variable and simplify
Isolating/rearranging: solve for one variable in terms of another (used when no value is given)
Isolating steps: move terms to isolate the variable’s term, then divide by its coefficient
- Example: $2 x + 3 y = 6$ solved for $y$ gives $y = \frac{- 2 x + 6}{3}$

Coefficients vs. exponents

Coefficient = number in front of a variable, shows how many of that variable (e.g., $3 x = x + x + x$ )
Exponent = small raised number showing repeated multiplication (e.g., $x^{3} = x \cdot x \cdot x$ )
Don’t confuse the two — they represent different operations

One-step equations

Solved by identifying the operation applied to the variable and undoing it with its inverse
Addition ↔ subtraction; multiplication ↔ division are inverse pairs
Always apply the same operation to both sides to keep the equation balanced

Substitution practice

Replace the variable with its given value, then simplify using order of operations
Example: if $x = 6$ , then $x^{2} - 17 = 36 - 17 = 19$

Next | 2.3.2 Rules and operations

Expressions and equations

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

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

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 equation becomes:

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

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)

Variables

Definitions

Polynomial: An expression with multiple terms that can’t be combined
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 $3 + 5 = 8$ .

Multiple variables

Sidenote

Important to know!

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:

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

Here is the full solution shown together:

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

Sidenote

Don't worry

If this process doesn’t look familiar, keep reading. We’ll show you exactly how it’s done!

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 $36 + 12 = 48$ .

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

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)

$12$