Combining like terms and clearing brackets so you can add and subtract expressions down to their simplest form.
Point 4 of 5 in this lesson: 7.3.4 Adding and subtracting expressions
Back in Lesson 7.2 you met the polynomial family — terms, coefficients, and the difference between a monomial and a polynomial. Now you will learn the two moves that let you actually work with those expressions: combining like terms (squeezing an expression down to fewest pieces) and clearing brackets (getting rid of the parentheses so the terms are free to combine). Put the two together and you can add or subtract any two expressions and write the answer in its simplest form.
By the end of this lesson you will be able to do five things: add like terms by adding only their coefficients; remove brackets correctly — letting a + sign pass terms through unchanged and treating a − sign as a mirror that flips every sign inside; put terms back into brackets in reverse; carry out a full add-or-subtract of two expressions; and use all of it on real problems like the perimeter of a shape with algebraic sides. We keep one steady habit of color: the letter is blue, the number in front is amber, and a flipped sign turns red.
Start with something you have known since you were small. If you have 3 apples and someone hands you 2 apples, you have 5 apples. You added the counts — 3 and 2 — and the thing being counted, "apples," did not change. Algebra works the very same way, except the thing being counted is a letter.
A term like 3x means "three of the thing called x." The amber number in front is the coefficient — it counts how many. So 3x + 2x is "three x's plus two x's," which is five x's: 3x + 2x = 5x. You added the coefficients and kept the letter part exactly as it was.
Two terms can be combined only when their letter part is identical — same letter, same exponent. We call such terms like terms. The squared term x2 behaves the same way: 4x2 − x2 = 3x2, because "four of the x2-things take away one of them leaves three." (Notice a bare x2 means 1x2 — the coefficient 1 is just left unwritten.)
But x and x2 are not like terms. One counts x's, the other counts x2's — different things entirely, the way apples and oranges are different. You can no more turn x + x2 into a single term than you can call "one apple and one orange" a tidy "two of something." It just stays x + x2.
To combine like terms, add (or subtract) only the coefficients and keep the letter part unchanged. Terms can combine only if their letter-and-exponent part matches exactly. 3x + 2x = 5x; 4x2 − x2 = 3x2; x + x2 stays as it is.
Simplify 5x + 3 + 2x − 1.
Do not merge unlike terms. 3x + 4 is not 7x — the 4 is not counting x's, so it has to stay separate. And never combine x with x2; the exponent makes them different things.
Add tiles to three bins — x2, x, and plain units. Inside a bin the coefficients add; the bins never mix, because they hold different things. Build 3x + 2x = 5x and watch x and x2 stay apart.
Brackets group terms together, but to combine like terms you usually need those terms set free. How you remove a bracket depends on one thing: the sign sitting in front of it.
A + sign in front is a friendly door. Every term inside walks out exactly as it was, signs and all. So a + (b − c + d) = a + b − c + d. Nothing changes; the brackets simply disappear.
A − sign in front is a mirror. A minus reverses everything it touches, so every term that comes out through it has its sign flipped: a plus becomes a minus, a minus becomes a plus. This is the single most important skill in the whole lesson, so look closely:
Why a mirror? Because −(b − c + d) really means "subtract the whole bundle." Subtracting a bundle is the same as subtracting each piece of it, and subtracting a piece reverses its sign. (If you have met multiplying out, this is exactly multiplying the bracket by −1.) Either way, the rule is short: a minus in front flips every sign inside.
+ ( ⋯ ) → copy every term unchanged. − ( ⋯ ) → flip the sign of every term inside. The first term inside, if it has no written sign, counts as a +.
Remove the brackets in 8 − (3 − 2y + y2).
The most common slip in all of algebra: flipping only the first term after a minus. a − (b − c) is not a − b − c. The mirror touches everything inside, so it is a − b + c.
Flip the sign in front of the bracket. With a +, the terms pass straight through. With a −, watch each term cross the mirror and flip its sign. The result line updates live.
Adding brackets is just removing brackets run backwards: you take some loose terms and tuck them into a bracket. This is handy for grouping, and — even better — it gives you a way to check your bracket-removal, since wrapping the terms back up should hand you exactly what you started with.
The rule mirrors the one before. If you put a + in front of the new bracket, the terms go in unchanged. If you put a − in front, then every term you tuck inside must flip its sign on the way in — because that leading minus will flip them right back when the bracket is later opened.
Tuck the last two terms of 5x − 2y + 3 into a bracket with a − in front.
Choose how many trailing terms to tuck into a bracket and pick a leading + or −. The bracketed form is built for you, with sign flips applied when the lead is −, and the widget checks it equals the original.
Now everything comes together. To add or subtract two expressions, follow two steps in order: (1) clear the brackets (remember — a minus flips every sign inside), then (2) combine the like terms. That is the whole procedure.
Take (3x + 2) − (x − 5). The first bracket has a + in front (just write the terms), the second has a − in front (flip everything inside). Clearing gives 3x + 2 − x + 5. Now gather like terms: the x-terms 3x − x = 2x, and the numbers 2 + 5 = 7. The answer is 2x + 7.
Simplify (3x + 2) − (x − 5).
When subtracting, the minus belongs to the whole second bracket. Clear the brackets before you start combining — try to combine first and you will almost always forget to flip a sign hiding inside.
Pick a problem, then press Next step to reveal the work: first clear the brackets, then combine like terms, then read off the result. Every step is exact.
These two moves earn their keep the moment a real measurement is unknown. Suppose a rectangle is 2x + 1 units wide and x units tall. You do not know x yet, but you can still write its perimeter — the distance all the way around — as one tidy expression. A rectangle has two widths and two heights, so add them up and combine like terms.
Find the perimeter of a rectangle with width 2x + 1 and height x.
Subtraction shows up just as naturally in a price difference. Say a hardcover book costs 3p + 5 dollars and the paperback costs p + 2 dollars. How much more is the hardcover? Subtract: (3p + 5) − (p + 2). The minus flips the second bracket: 3p + 5 − p − 2 = 2p + 3. The hardcover costs 2p + 3 dollars more, whatever p turns out to be.
Whenever a quantity is built from parts you do not have numbers for yet, write each part as an expression, then add or subtract the parts and simplify into one clean expression. Perimeters, totals, and differences all follow the same clear-then-combine routine.
Choose a shape and slide its side expressions. The widget adds the sides and simplifies the perimeter live — and it is always algebraically correct.
To combine like terms, add only the coefficients and keep the letter part the same — and only terms with an identical letter-and-exponent part can combine. To remove brackets, a + in front lets every term out unchanged while a − in front is a mirror that flips every sign inside: a − (b − c + d) = a − b + c − d. Adding brackets runs that backwards, flipping each term you tuck behind a minus. And to add or subtract two expressions, clear the brackets first, then combine like terms — exactly how (3x + 2) − (x − 5) = 2x + 7.
Work each one out first, then open the answer to check your thinking.
Six questions to lock it in. Tap the answer you think is right.
You just learned to add and subtract whole expressions by clearing brackets and combining like terms. That built on the term-and-coefficient ideas from 7.2 The Polynomial Family. Coming up in 7.4 Working with Powers, you will see what happens to the exponents when terms are multiplied rather than added — the next big move in algebra.