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Learn to add, subtract, factor, and expand expressions with fractions and decimals — the building blocks of algebra.
Have you ever wondered why we can rearrange numbers in a math problem and still get the same answer? People have been noticing patterns like this for thousands of years. The properties of operations — like the commutative, associative, and distributive properties — are rules that mathematicians discovered by studying how numbers behave. These rules didn't appear in a textbook overnight. They were built up over centuries of careful thinking.
x or =.3x + 2 instead of describing it in words made algebra much faster and clearer.Today, you use these same properties every time you simplify an expression. You're building on ideas that are thousands of years old!
Before we dive into examples, let's make sure you know the key vocabulary and the four big properties you'll use. A linear expression is a math phrase that contains variables (like x or y) raised only to the first power, combined with numbers using addition, subtraction, and multiplication. A rational coefficient is simply a number in front of a variable that can be written as a fraction — this includes whole numbers, decimals, and fractions like ¾ or −2.5.
a + b = b + a and a × b = b × a.(a + b) + c = a + (b + c). The grouping changes, but the answer doesn't.a(b + c) = ab + ac. This is the key to expanding expressions.3x and 5x) can be added or subtracted. You just work with their coefficients: 3x + 5x = 8x.6x + 9 = 3(2x + 3). It's the opposite of expanding!Let's look at how the distributive property works with a picture. Imagine you have a rectangle whose width is 3 and whose length is split into two parts: x and 4. The total area is 3(x + 4). But you can also find the area of each smaller rectangle and add them together: 3x + 12. Both methods give the same total area — that's the distributive property in action!
Now let's look at combining like terms visually. When you see an expression like 2x + 3 + 5x − 1, think of sorting items into groups: all the x-terms go together, and all the plain numbers (called constants) go together.
Notice how the colors help you see what belongs together. The cyan terms are the variable terms, and the amber terms are the constants. Sorting and combining like terms is one of the most important skills you'll practice.
Let's lay out the main operations you'll perform on linear expressions. Remember, a linear expression looks something like ¾x − 2.5y + 7. The numbers ¾ and −2.5 are the rational coefficients (the numbers stuck to the variables).
When you add or subtract two expressions, you combine like terms. Like terms have the exact same variable part. Here's the general pattern:
This works with fractions and decimals too. For example, ½x + ¾x = (½ + ¾)x = ⁵⁄₄x. You just add the fractions like you normally would, then attach the x.
When a number sits outside parentheses, you multiply it by every term inside. This is called expanding or distributing.
Factoring is the reverse of distributing. You find a number or variable that divides evenly into every term, and you pull it out in front of parentheses.
For example, in 10x + 15, both terms share a factor of 5. So 10x + 15 = 5(2x + 3). You can always check by distributing the 5 back in — you should get the original expression.
Let's break down each operation into clear steps you can follow every time. The table below shows the strategy for each type of problem, along with a quick example.
| Operation | Strategy | Example |
|---|---|---|
| Adding expressions | Group like terms, then add their coefficients. | (3x + 2) + (5x − 7) = 8x − 5 |
| Subtracting expressions | Distribute the negative sign to every term in the second expression, then combine like terms. | (4x + 6) − (x + 1) = 3x + 5 |
| Expanding | Multiply the outside number by each term inside the parentheses. | −2(3x − 4) = −6x + 8 |
| Factoring | Find the GCF of all terms. Write it outside parentheses with the "leftovers" inside. | 12x − 8 = 4(3x − 2) |
| Mixed (multi-step) | Expand first, then combine like terms. | 2(x + 3) + 4x = 6x + 6 |
Here's something that trips a lot of students up: subtracting an expression. When you see a minus sign before parentheses, like −(2x + 5), that minus sign means "multiply everything inside by −1." So it becomes −2x − 5. Both signs flip! This is the distributive property with a = −1.
Another common situation is working with fractions as coefficients. The same rules apply — you just need to be comfortable adding and multiplying fractions. For instance, to combine ⅓x + ½x, find a common denominator (6), rewrite as ²⁄₆x + ³⁄₆x, and add to get ⁵⁄₆x.
Let's work through a problem that uses several of our strategies together. Take your time and follow each step.
½ by each term inside: ½ × 4x = 2x and ½ × (−6) = −3.3 by each term inside: 3 × ⅓x = x (because 3 × ⅓ = 1) and 3 × 2 = 6.2x + x = 3x. Group the constants: −3 + 6 = 3.These properties are incredibly powerful — they work on every linear expression you'll ever see. But there are a few spots where students often slip up. Let's be honest about what to watch out for.
| ✓ Strengths | ✗ Common Pitfalls |
|---|---|
| Works with all rational numbers — integers, fractions, decimals. | Forgetting to distribute the negative sign to every term inside parentheses. |
| You can always check your work by reversing the operation (distribute to check factoring, factor to check distributing). | Combining unlike terms — for example, adding 3x and 5 to get 8x. (They're not like terms!) |
| The commutative and associative properties let you rearrange freely, which makes grouping easier. | Making errors with fraction arithmetic — especially finding common denominators when adding coefficients. |
| These same strategies carry forward into equations, inequalities, and higher math. | Forgetting that subtraction is the same as adding a negative. Writing 5 − 3x as 5 + (−3x) can help. |
The skills you're learning right now are the foundation of algebra. Everything builds on simplifying expressions. Here's a peek at how this topic connects to what's ahead.
| What You're Learning Now | Where It Leads |
|---|---|
| Combining like terms in expressions | Solving one-step and two-step equations (7th & 8th grade) |
| Distributing with rational coefficients | Solving multi-step equations and inequalities (8th grade) |
| Factoring linear expressions | Factoring quadratic expressions like x² + 5x + 6 (Algebra 1) |
| Writing equivalent expressions | Proving that two expressions are equal — algebraic proof (high school) |
In 8th grade, you'll start using these simplification skills to solve equations — finding the actual value of x that makes an equation true. For example, if 3(x + 1) = 12, you'd first expand to get 3x + 3 = 12, and then solve for x. See how expanding is the first step? That's exactly what you're practicing now.
In high school Algebra 1, you'll factor not just linear expressions but quadratics — expressions with x² in them. The factoring skills you build now will make that transition much smoother. Think of this lesson as training wheels that get you ready for bigger rides.
Try these five problems on your own. When you're ready, click "Show Answer" to check your work. Each problem builds on the one before it.
5(x + 3) as 5x + 15?7x + 3 − 4x + 9¾(8x − 12) + 2x$12.50 per hour babysitting plus a flat $8 travel fee. She worked for h hours on Saturday and h hours on Sunday, but on Sunday the family gave her a $5 tip. Write and simplify an expression for her total earnings over both days.⅓(6x + 6) − ⅔(3x − 3) simplifies to a constant (a number with no variable). What does this tell you about the original expression?In this lesson, you learned how to use the properties of operations — the commutative, associative, and distributive properties — as strategies for manipulating linear expressions with rational coefficients. You practiced four key skills: adding expressions by combining like terms, subtracting expressions by distributing the negative sign first, expanding (distributing a number across parentheses), and factoring (pulling out a common factor). Along the way, you saw how these ideas connect to thousands of years of mathematical history and lay the foundation for solving equations in algebra.
Remember: like terms must have the exact same variable part before you can combine them. Expanding and factoring are opposite operations — use one to check the other. And when working with fractions and decimals, the same rules apply — just take extra care with your arithmetic. These strategies are your toolkit for the rest of algebra. Keep practicing, and they'll become second nature!