Distributive Property and Combining Like Terms

To simplify an expression like 4(3x - 5) + 2x, multiply the 4 into every term inside the parentheses, then add terms that have the same variable part. That gives 12x - 20 + 2x = 14x - 20. Those two moves, distributing and combining like terms, appear in the grade 6 standards (6.EE.A.3) and carry through to solving equations in grade 8 (8.EE.C.7.b).

1. The distributive property

A factor outside parentheses multiplies each term inside. The grade 6 standard gives the example 3(2 + x) = 6 + 3x. In symbols, a(b + c) = ab + ac.

It also runs backward. The standard shows 24x + 18y = 6(4x + 3y), which pulls out the greatest common factor, 6. This is factoring, and it is the first step in many later topics.

2. Like terms

Like terms have the same variable part. 5x and 3x are like terms. 5x and 5 are not, and neither are 5x and 5x^2, because x and x^2 are different quantities.

To combine like terms, add the coefficients and keep the variable part: 5x + 3x = 8x. The standard puts it as "y + y + y equals 3y," and two expressions are equivalent when they name the same number no matter which value the variable takes (6.EE.A.4).

3. Example 1: distribute, then combine

Simplify 4(3x - 5) + 2x.

1. Distribute: 4 x 3x = 12x, and 4 x (-5) = -20. You now have 12x - 20 + 2x.
2. Combine like terms: 12x + 2x = 14x.
3. Result: 14x - 20.

4. Example 2: a negative factor

Simplify -2(x - 7).

Multiply -2 by each term: -2 x x = -2x, and -2 x (-7) = +14.

Result: -2x + 14.

The most frequent slip is to write -2x - 14. The sign of the second term flips because a negative times a negative is positive.

5. Example 3: subtracting a whole group

Simplify 3(2x + 1) - (x - 4).

1. Distribute 3: 6x + 3.
2. The minus sign in front of (x - 4) acts like multiplying by -1: -x + 4.
3. Put it together: 6x + 3 - x + 4.
4. Combine x terms: 6x - x = 5x. Combine numbers: 3 + 4 = 7.
5. Result: 5x + 7.

If you forget to change the sign of the 4, you get 5x - 1, which is wrong.

6. Check by substitution

Pick any value for x and see whether the original and your answer agree. This is the same test of equivalence the standards describe.

Let x = 2 in Example 3:

• Original: 3(2 x 2 + 1) - (2 - 4) = 3(5) - (-2) = 15 + 2 = 17
• Answer: 5(2) + 7 = 17

They match. In Example 1 with x = 3, the original is 4(9 - 5) + 6 = 22, and 14(3) - 20 = 22. Also a match.

One value that matches is strong evidence but not proof, so use a second value if the problem looks tricky. Avoid 0 and 1 as test values, since many errors still pass with them.

7. Example 4: use it to solve an equation

Solve 2(x + 3) + 4x = 30.

1. Distribute: 2x + 6 + 4x = 30
2. Combine: 6x + 6 = 30
3. Subtract 6 from both sides: 6x = 24
4. Divide by 6: x = 4

Check: 2(4 + 3) + 4(4) = 14 + 16 = 30. It works.

The grade 8 standard names this directly: solve linear equations "whose solutions require expanding expressions using the distributive property and collecting like terms."

8. A word problem

A school play charges $6 for a child ticket and $9 for an adult ticket, plus a $2 fee per order. A family buys 3 child tickets and 2 adult tickets. The total is 6c + 9a + 2 with c = 3 and a = 2: 18 + 18 + 2 = $38.

Distribution helps when a group buys the same bundle. If each of n people buys a $6 ticket and a $3 snack, the cost is n(6 + 3) = 9n, so every person adds $9.

9. Common mistakes

• Distributing to only the first term. 3(x + 4) is 3x + 12, not 3x + 4.
• Dropping a negative. A minus sign before parentheses changes every sign inside.
• Combining unlike terms. 3x + 2x^2 cannot be written as 5x^3 or 5x^2.
• Adding exponents when combining. 3x + 2x = 5x, not 5x^2.

10. Key takeaways

• a(b + c) = ab + ac: multiply every term inside the parentheses.
• Like terms have identical variable parts; add the coefficients only.
• A minus sign before parentheses flips every sign inside.
• Substitute a number into both versions to check they agree.
• Distribute and collect terms first, then solve the equation.