Algebra 1
Combining like terms & the distributive property
Before you can solve a messier equation, you have to tidy it: gather the pieces that belong together and clear away the parentheses. That's all this lesson is, housekeeping for expressions.
Two of the moves here, the area picture especially, come back in later units, so they're worth getting comfortable with now. And one of them, distributing a negative, is the place a sign goes missing more than anywhere else in algebra, so it gets extra care.
Start with combining, using the picture of a box that holds a hidden number. Picture 3x as three identical boxes, each holding the same hidden number, and 2x as two more of that same box. Put them together and you have five of that box: 3x + 2x = 5x. You're just counting boxes.
But 3x + 2 is different. That's three boxes plus two loose coins. There's no way to fold loose coins into boxes and call it one thing, so 3x + 2 simply stays as it is. Only matching kinds combine.
Now distributing. The number outside a set of parentheses has to reach everyone inside. Picture handing a flyer to every person in a room, skipping no one. So in 2(x + 4), the 2 greets the x and the 4 both. A clean way to see why is to draw it as the area of a rectangle, 2 tall and (x + 4) wide, split into two cells: $$\begin{array}{c|c|c} & x & 4 \\ \hline 2 & 2x & 8 \end{array}\qquad\Rightarrow\qquad 2(x+4)=2x+8$$ The two cells are the two products, 2x and 8, and together they're the whole area: 2x + 8.
Then the one that bites. When the thing out front is a minus sign, it's really a −1, and it shakes hands with every term inside, including their signs. Take −(x − 4). The −1 meets the x and gives −x. Then it meets the −4, and −1 times −4 is +4. So −(x − 4) is −x + 4.
Substitute x = 1 to see it: the original is −(1 − 4) = −(−3) = 3, and the rewrite −1 + 4 is also 3. They agree.
New words
- 2.3.d1 Term: a number, a variable, or numbers and variables multiplied together; in an expression the terms are the parts being added. A minus sign belongs to the term that follows it (subtracting is adding a negative), so in 3x − 5 the terms are 3x and −5. The sign travels with its term. That's exactly why distributing −2 over (x − 5) has to send −2 to both x and −5: the −5 is a term, sign and all.
- 2.3.d2 Coefficient: the number multiplying the variable (the 3 in 3x).
- 2.3.d3 Like terms: terms with the same variable part, the same variable(s) raised to the same power. Right now that just means the same variable: 3x and 2x are like; 3x and 2 are not (one is "boxes," the other is "loose units"). Later, 3x² and 5x² will be like, but 3x and 3x² will not.
- 2.3.d4 Distributive property: a(b+c)=ab+ac, and likewise a(b−c)=ab−ac. Subtracting is adding a negative, so it's the same rule. Just keep the sign. The outside factor multiplies everything inside.
Worked example
2.3.w1 Combine like terms: $$3x+2x=5x$$
2.3.w2 Combine, leaving unlike terms alone: $$7x-4x+2 = 3x+2 \quad(\text{the }+2\text{ has no like partner})$$
2.3.w3 Distribute (positive): $$2(x+4)=2x+8$$
2.3.w4 Distribute a negative, the big sign trap: $$-(x-4) = -x+4 \qquad \text{Check at }x=1:\; -(1-4)=3 \;\text{ and }\; -1+4=3$$
2.3.w5 Distribute a negative, then combine (the trap in full): $$3-2(x-5) \;=\; 3 + (-2)(x) + (-2)(-5) \;=\; 3 - 2x + 10 \;=\; -2x+13$$ Walk the last one slowly, because it's the whole skill in one line. The −2 reaches both terms inside: −2 times x is −2x, and −2 times −5 is +10 (a negative times a negative). Then 3 + 10 is 13, leaving −2x + 13. Check at x = 1: the original is 3 − 2(1 − 5) = 3 − 2(−4) = 3 + 8 = 11, and the rewrite −2(1) + 13 is also 11.
After one clean run, here's the slip to know about. The tempting wrong answer is −2x − 7, which comes from giving −2 × −5 a minus sign and writing −10. But two negatives multiply to a positive, so that product is +10, not −10. A quick self-check catches it every time: substitute x = 1 into both your answer and the original, and they'll agree only if the sign is right.
Check yourself
- 2.3.c1 Simplify 5x + 2 − x. Which terms are "like," and which loner stays put? (5x and −x are like, since five boxes minus one box is 4x, and the +2 is loose, so the answer is 4x + 2.)
- 2.3.c2 Expand −(2x + 5) and confirm the signs by testing x = 1. (The −1 meets both terms: −2x and −5, so −(2x + 5) = −2x − 5. Check at x = 1: original −(7) = −7, rewrite −2 − 5 = −7.)
- 2.3.c3 Someone writes 4 − 2(x + 3) = 4 − 2x + 6. Where exactly did it go wrong? (The −2 reaches the +3 as −2 × 3 = −6, not +6, so the correct line is 4 − 2x − 6, which simplifies to −2x − 2.)
You can now combine like terms by counting matching boxes, distribute a number across parentheses so it reaches every term, and handle a leading negative without dropping its sign, checking with a quick substitution when a sign is in doubt.
The set below is deliberately mixed, and several problems lead with a negative so the sign work stays sharp.
Combine like terms:
Reveal answerHide to problem 1
7xReveal answerHide to problem 2
7xReveal answerHide to problem 3
4x+2Reveal answerHide to problem 4
4x+2Reveal answerHide to problem 5
4xDistribute:
Reveal answerHide to problem 6
3x+6Reveal answerHide to problem 7
5x-15Reveal answerHide to problem 8
8x+4Distribute a leading negative:
Reveal answerHide to problem 9
-x+7Reveal answerHide to problem 10
-2x-5Distribute, then combine: