SAT Combining Like Terms and Solving Equations: Lesson, Practice Problems & Video Explanations

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By Glyn Caddell, BSME · Author of Digital SAT Math Secrets · Tutoring since 2008 — Updated August 18, 2026

Combining like terms & solving equations appears approximately 11.7% of the time on the Digital SAT. These questions fall under two different categories according to the College Board. Combining like terms falls under Advanced Math, and solving equations falls under Algebra. Combining like terms & solving equations questions test students’ ability to combine like terms and solve simple or complex equations, but in many cases you can use the Desmos calculator to solve them.

Table of Contents

Lesson
- Combining Like Terms Lesson
- Solving Equations Lesson
  - How to Solve Equations with Desmos
  - How to Solve Equations Algebraically
Tips
Example
- Video Explanation
- Text Explanation
Practice Problems with Video Explanations
Common Mistakes
FAQs
Related Topics

Combining Like Terms & Solving Equations Lesson

The lesson covers the types of Combining Like Terms & Solving Equations questions that show up on the Digital SAT, how to approach the questions, and an example of an SAT question.

What Do Combining Like Terms & Solving Equations Questions Look Like?

Combining Like Terms & Solving Equations questions typically have an expression that can be simplified or an equation that can be solved for a single variable. Here are some examples of Combining Like Terms & Solving Equations questions from the Digital SAT.

What is the solution to the given equation?
What value of $x$ is the solution to the given equation?
Which expression is equivalent to…

1. Combining Like Terms

What are Like Terms?

Like terms have:

The same variable
Raised to the same exponents

Only the coefficients (the numbers in front of the terms) may be different.

Examples of like terms

$3x$ and $-5x$
$12x^2$ and $x^2$
$-3xy$ and $-7xy$
$12x^2y$ and $9x^2y$
$13a$ and $2a$

Examples of unlike terms

$3x$ and $-4y$
$12x^2$ and $2x$
$-3xy$ and $-7xz$
$11x^2y$ and $2xy^2$
$3a$ and $21b$

How to Combine Like Terms

Identify like terms
- Group terms with the same variable(s) and exponents
Combine them by adding coefficients
- Leave the variable part unchanged
Watch for signs in front of each term
- A minus sign is really a negative sign

Example

Simplify the following expression.

$3x-2xy+3y^2-4x+12xy+2y^2$

Group Like Terms and Combine

$x$ -terms:

$3x + (-4x)$ , to get

$-x$

$xy$ -terms:

$-2xy+12xy$ , to get

$10xy$

$y^2$ -terms:

$3y^2+2y^2$ , to get

$5y^2$

Final Answer

$-x+10xy+5y^2$

Adding and Subtracting Expressions

When questions ask us to add and subtract expressions, we are really being asked to combine like terms.

Example

What is the sum of $x^2-2x+7$ and $2x^2-3x-4$ ?

Explanation & Solution

To solve this let’s add them together.
$(x^2-2x+7)+(2x^2-3x-4)$
Distribute the plus sign $(+1)$ to each of the terms in the second expression. When you do that, all of the terms remain the same and we can drop the parentheses.
$x^2-2x+7+2x^2-3x-4$
Now, just combine like terms to get
$3x^2-5x+3$

Example

Simplify $(4x^2-5x+2)-(2x^2+x-3)$

Explanation & Solution

Distribute the minus sign $(-1)$ to each of the terms in the second expression. When you do that, all of the terms get negated and we can drop the parentheses.
$4x^2-5x+2-2x^2-x+3$
Now, just combine like terms to get
$2x^2-6x+5$

2. Solving Equations

To solve equations on the Digital SAT students can solve them using Desmos or solve them algebraically.

How to Solve Equations with Desmos

Most equations can be solved using the Desmos calculator.

The graph of $x=5$ will be a vertical line passing through the $x$ -axis at 5.

The reason is that every point on that line has an $x$ -value of 5. The $y$ -value changes, but $x$ remains 5.

This is a graph of $x=5$ in Desmos.

We can use this to solve equations with one variable. Simply, type in the equation into Desmos.

For example, the solution to $\dfrac{2}{3} x+5=x+2$ is 9.

That means the equation $\dfrac{2}{3} x+5=x+2$ is equivalent to $x=9$ . Therefore, if we graph the equation $\dfrac{2}{3} x+5=x+2$ , we will just get the graph of $x=9$ .

We can solve $\dfrac{2}{3} x+5=x+2$ just by graphing it and checking the $x$ -intercept.

Solving Equations with Square Roots or Absolute Values

The same method of using Desmos can be used to solve equations with square roots or absolute values. Make sure to zoom out when using Desmos because you may get two solutions, but only one might be visible in the default view on Desmos. Also, note that you cannot click on the $x$ -intercepts when the solutions to square root or absolute value equations are graphed. Zoom in to make sure the solution is the value you expect.

Example of Solving an Equation with a Square Root Using Desmos

Example of Solving an Absolute Value Equation Using Desmos

How to Solve Equations Algebraically

To solve equations algebraically, you need to use inverse operations and apply order of operations (PEMDAS) in reverse to isolate the variable you are solving for.

For example, let’s examine the equation

$3x+2=23$

$x$ represents a number, so if we treat $x$ like a number and think about how we would follow order of operations on the left side of the equation, we would

Multiply 3 and $x$
Add the 2

To solve the equation we have to do order of operations in reverse and use inverse operations.

Adding the 2 is done last in order of operations, so we would take care of that first. We have to do inverse operations, so we would subtract 2.

Multiplying by 3 is done first in order of operations, so we would take care of that last. We have to do inverse operations, so we would divide by 3.

Subtract 2 from both sides
Divide both sides by 3

Here are the steps solving the equation:

$3x+2=23$

subtract 2 from both sides

$3x=21$

divide both sides by 3

$x=7$

Some tricks to make solving equations easier include choosing to distribute or divide a coefficient, multiplying all terms by the LCD to eliminate fractions, and cross multiplying when a fraction is set equal to a fraction.

Distribute a Coefficient Into Parentheses or Divide by the Coefficient

If there is a number outside a set of parentheses, you can choose to distribute it or divide it.

The equation $3(x-4)=15$ can be solved two different ways.

We can either distribute the $3$ to each of the terms or divide by the $3$ .

Example of Distributing

$3(x-4)=15$

distribute the $3$ into the parentheses and the equation becomes

$3x-12=15$

add $12$ to both sides of the equation

$3x=27$

divide both sides by $3$ to get

$x=9$

Example of Dividing First

$3(x-4)=15$

divide both sides by $3$ to get

$x-4=5$

add $4$ to both sides of the equation to get

$x=9$

With both methods we are able to get $x=9$ . However, on the SAT there may be questions in which one method works better than another.

Example

$9(2x+3)=36$ . What is the value of $2x+3$ ?

Explanation & Solution

In this question, the expression we are solving for can be found directly.

If we divide both sides by $9$ , we will have the answer immediately. It would be a waste of time to distribute the $9$ .

$9(2x+3)=36$

divide both sides by $9$ to get

$2x+3=4$

The answer is $4$ .

Multiply All Terms by the Least Common Denominator (LCD) to Eliminate Fractions in Equations

The following equation

$\dfrac{2}{3}x-\dfrac{4}{5}=\dfrac{1}{6}$

can be intimidating for students.

Many students are not comfortable working with fractions. Plus, there is room to make mistakes with fractions.

We can multiply all of the terms in the equation by the LCD to cancel out the denominators, so we are left with integers.

Example

What is the value of $x$ in the following equation?
$\dfrac{2}{3}x-\dfrac{4}{5}=\dfrac{1}{6}$

Explanation & Solution

$\dfrac{2}{3}x-\dfrac{4}{5}=\dfrac{1}{6}$
The denominators in the equation above are 3, 5, and 6.
To find the least common denominator, we need the least common multiple of the denominators.

In this case, the LCD is $30$ .

We should multiply each term in the equation by 30 to get
$(30)\dfrac{2}{3}x-(30)\dfrac{4}{5}=(30)\dfrac{1}{6}$

Reduce each term to get

$(10)2x-(6)4=(5)1$

Simplifying further gives us

$20x-24=5$

Now, we can solve like normal.

Add $24$ to both sides to get

$20x=29$

Divide both sides by $20$ to get

$x=\dfrac{29}{20}$

Cross Multiply When a Fraction Equals a Fraction

If you have an equation in which one fraction equals another fraction, it is a proportion. To solve a proportion you should cross-multiply.

It is best to simplify each fraction as much as possible before cross-multiplying.

Example

Solve for $x$ in the equation below.

$\dfrac{2x-7}{4}=\dfrac{2x-1}{12}$

Explanation & Solution

$\dfrac{2x-7}{4}=\dfrac{2x-1}{12}$

Cross-multiply to get

$(2x-7)(12)=(2x-1)(4)$

Simplify both sides

$24x-84=8x-4$

Since there is an $x$ -term on both sides of the equation, we should get them both on one side.

Subtract $8x$ from both sides. (You can subtract $24x$ from both sides instead. It is up to you)

$16x-84=-4$

Add $84$ to both sides to get

$16x=80$

Divide both sides by $16$

$x=\dfrac{80}{16}$

Reduce

$x=5$

Tips

Distribute leading coefficients including their signs to each term within parentheses.
Identify like terms before combining them. Combine like terms by adding them together, treating a leading minus sign as a negative sign.
Practice solving equations using Desmos, so you can answer the questions quickly on the Digital SAT.
Also, practice solving equations by hand because the skill is necessary for solving for a variable in terms of another.
Double check the prompt to identify exactly what the question is asking for: is it asking to solve for just $x$ or for something else such as $x+3$ ?

Example

If $3(2x-5)-4(x+1)=17$ , what is the value of $x+8$ ?

A) 18
B) 24
C) 26
D) 27

Video Explanation

Text Explanation

Answer: C

First distribute:

$3(2x-5)=6x-15$
$-4(x+1)=-4x-4$

So the equation becomes:

$6x-15-4x-4=17$

Combine like terms:

$2x-19=17$

Add 19 to both sides:

$2x=36$

Divide by 2:

$x=18$

Now find $x+8$ :

$18+8=26$

Therefore, the correct answer is C.

Common Mistakes

Three common mistakes students make when answering Combining Like Terms & Solving Equations questions are dropping the negative sign, mixing up exponent rules, and answering the wrong prompt.

Dropping the Negative Sign

A common mistake for combining like terms and for solving equations is dropping the negative sign. One instance this happens often in is when there is a minus sign on the outside of a parenthesis. A way to avoid the mistake is to distribute the minus sign as a negative to each term in the parenthesis.

For example,

$-(x-5) = -x+5$

Students may accidentally do the following:

$-(x-5)=-x-5$ , which is wrong.

Mixing Up Exponent Rules

A common mistake is to add the exponents when combining like terms. Exponents should only be added when terms with the same bases are multiplied. When they are added, the coefficients should be added if they are like terms, and the exponents should remain the same.

For example,

$x^3+x^3=2x^3$

A common mistake would be to add the exponents like the following:

$x^3+x^3=x^6$ , which is wrong.

Answering the Wrong Prompt

Another common mistake is answering the wrong prompt. A question may ask for the value of $x+3$ , but the student answers the question as if it was asking for the value of $x$ .

Frequently Asked Questions

Here are three frequently asked questions that I get from students when reviewing Combining Like Terms & Solving Equations.

How do I solve equations with the Desmos calculator?

To solve a single-variable equation using Desmos, simply enter the equation as it is into Desmos. If there is an answer it will be a vertical line through the $x$ -coordinate of the solution. For example, if the answer is 5, Desmos will display a vertical line through the $x$ -axis at $x=5$ .

Note that if the equation has two solutions, the graph will display two vertical lines.

How do I solve for an unknown coefficient or constant, k?

To solve for an unknown coefficient or constant, $k$ , simplify an expression until it matches the format of the expression in the question. Then match up all of the terms with the terms in the question to identify $k$ .

For example, a question may state that a given expression equals $kx+7$ and ask for the value of $k$ .

For this question keep on simplifying an expression until it is in the format $kx+7$ . If you end up with $3x+7$ , then $k=3$ .

What kind of equations can be solved using Desmos?

Desmos can be used to solve almost any single-variable equation including linear equations, quadratic equations, exponential equations, equations with square roots, and equations with absolute values.

SAT Combining Like Terms and Solving Equations: Lesson, Practice Problems & Video Explanations

Combining Like Terms & Solving Equations Lesson

What Do Combining Like Terms & Solving Equations Questions Look Like?

1. Combining Like Terms

What are Like Terms?

How to Combine Like Terms

Example

Group Like Terms and Combine

Final Answer

Adding and Subtracting Expressions

Example

Explanation & Solution

Example

Explanation & Solution

2. Solving Equations

How to Solve Equations with Desmos

Solving Equations with Square Roots or Absolute Values

How to Solve Equations Algebraically

Distribute a Coefficient Into Parentheses or Divide by the Coefficient

Multiply All Terms by the Least Common Denominator (LCD) to Eliminate Fractions in Equations

Cross Multiply When a Fraction Equals a Fraction

Tips

Example

Video Explanation

Text Explanation

Practice Problems with Video Explanations

Common Mistakes

Dropping the Negative Sign

Mixing Up Exponent Rules

Answering the Wrong Prompt

Frequently Asked Questions

How do I solve equations with the Desmos calculator?

How do I solve for an unknown coefficient or constant, k?

What kind of equations can be solved using Desmos?

Related Topics

Solving for a Variable in Terms of Another

Linear Functions

All Math Topics