Simplifying Algebraic Expressions

Once you start working with variables, expressions can get messy pretty quickly. You might see something like \(3x + 5 - 2x + 7\) and wonder where to even begin. The good news? Simplifying expressions is just about recognizing patterns and applying a few basic rules. Think of it like cleaning your room—you're organizing similar items together and getting rid of clutter.

When we simplify an expression, we're not solving for a variable. We're just rewriting the expression in its cleanest, most compact form. This makes it easier to work with later when we do need to solve equations.

What Are Like Terms?

Like terms are terms that have the exact same variable part. The coefficients (the numbers in front) can be different, but the variable part must match exactly.

For example, \(3x\) and \(5x\) are like terms because they both have just \(x\). You can combine them: \(3x + 5x = 8x\). Think of it this way: if you have 3 apples and someone gives you 5 more apples, you now have 8 apples. Same logic.

But \(3x\) and \(3x^2\) are NOT like terms. One has \(x\) and the other has \(x^2\), which are completely different. You can't add 3 apples to 3 apple pies and get 6 of something—they're different things entirely.

Here are more examples of like terms:

  • \(7y\) and \(-2y\) are like terms (both have \(y\))
  • \(4ab\) and \(9ab\) are like terms (both have \(ab\))
  • \(x^2\) and \(-5x^2\) are like terms (both have \(x^2\))

And here are terms that are NOT alike:

  • \(5x\) and \(5y\) (different variables)
  • \(3a\) and \(3ab\) (different variable parts)
  • \(2x^2\) and \(2x^3\) (different exponents)

The constant terms (numbers without variables) are also like terms with each other. So \(5\), \(-3\), and \(12\) can all be combined together.

Combining Like Terms

Once you've identified like terms, combining them is straightforward—just add or subtract their coefficients.

Example: Simplify \(4x + 7x\).

Both terms have \(x\), so they're like terms. Add the coefficients: \(4 + 7 = 11\).

Answer: \(11x\)

Example: Simplify \(9y - 5y\).

Both terms have \(y\). Subtract the coefficients: \(9 - 5 = 4\).

Answer: \(4y\)

Now let's try something with multiple types of terms.

Example: Simplify \(5x + 3y - 2x + 8y\).

First, identify the like terms. We have \(5x\) and \(-2x\) (both have \(x\)), and we have \(3y\) and \(8y\) (both have \(y\)).

Combine the \(x\) terms: \(5x - 2x = 3x\)

Combine the \(y\) terms: \(3y + 8y = 11y\)

Answer: \(3x + 11y\)

Notice that \(3x\) and \(11y\) can't be combined further because they have different variables. We've simplified as much as possible.

Example: Simplify \(6a + 4 - 2a + 9\).

The like terms here are \(6a\) and \(-2a\), plus the constants \(4\) and \(9\).

Combine the \(a\) terms: \(6a - 2a = 4a\)

Combine the constants: \(4 + 9 = 13\)

Answer: \(4a + 13\)

Using the Distributive Property First

Sometimes you need to use the distributive property before you can combine like terms. Remember, the distributive property says \(a(b + c) = ab + ac\).

Example: Simplify \(3(x + 4) + 2x\).

First, distribute the 3: $$3(x + 4) = 3x + 12$$

Now we have: \(3x + 12 + 2x\)

Combine like terms: \(3x + 2x = 5x\)

Answer: \(5x + 12\)

Example: Simplify \(5(2y - 3) - 4(y + 1)\).

Distribute the 5: \(5(2y - 3) = 10y - 15\)

Distribute the -4: \(-4(y + 1) = -4y - 4\)

Now we have: \(10y - 15 - 4y - 4\)

Combine the \(y\) terms: \(10y - 4y = 6y\)

Combine the constants: \(-15 - 4 = -19\)

Answer: \(6y - 19\)

Pay special attention when distributing negative numbers. Each term inside the parentheses gets multiplied by that negative, which can flip signs in ways that trip people up.

A More Complex Example

Let's put everything together with a longer expression.

Example: Simplify \(7x + 3(2x - 5) - 4x + 8\).

Step 1: Distribute the 3. $$3(2x - 5) = 6x - 15$$

Step 2: Rewrite the expression. $$7x + 6x - 15 - 4x + 8$$

Step 3: Group like terms (this is just mental organization—you don't have to write this step). $$(7x + 6x - 4x) + (-15 + 8)$$

Step 4: Combine like terms. $$9x - 7$$

Answer: \(9x - 7\)

Why Simplifying Matters

You might wonder why we bother simplifying expressions if we're not actually solving for anything. Here's why it matters: simpler expressions are easier to work with in every way. They're easier to evaluate if you substitute a number for the variable. They're easier to solve if they're part of an equation. They're easier to graph. They're easier to understand.

Plus, many problems require simplification as a first step. If you're solving \(3(x + 2) + 4x = 26\), you can't solve it until you simplify the left side first.

Try These

  1. Simplify \(8x + 5x\)
  2. Simplify \(12y - 7y + 3\)
  3. Simplify \(4a + 6b - a + 2b\)
  4. Simplify \(2(x + 3) + 5x\)
  5. Simplify \(6(m - 2) - 3(m + 1)\)
  6. Simplify \(5x + 3 - 2x + 7 + x\)

Solutions:

  1. \(13x\)
  2. \(5y + 3\)
  3. \(3a + 8b\)
  4. First distribute: \(2x + 6 + 5x\), then combine: \(7x + 6\)
  5. Distribute: \(6m - 12 - 3m - 3\), then combine: \(3m - 15\)
  6. Combine \(x\) terms: \(5x - 2x + x = 4x\). Combine constants: \(3 + 7 = 10\). Answer: \(4x + 10\)

Watch Out For These Common Errors

Don't combine terms that aren't alike. You can't turn \(3x + 2y\) into \(5xy\) or anything else. If the variable parts don't match, leave them separate.

When distributing a negative, remember it affects ALL terms inside the parentheses. The expression \(-2(x - 3)\) becomes \(-2x + 6\), not \(-2x - 6\). That negative sign in front changes the \(-3\) into \(+6\).

Don't lose track of negative coefficients. If you have \(5x - 8x\), the answer is \(-3x\), not \(3x\). The subtraction matters.

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