➕ Algebra 1 · Equations
Combining Like Terms: fun ways and memory help
The same idea, the fun way
Pick the one that clicks. Each one explains the same concept from a different angle.
Hangouts by friend
Last week you hung out with Maya 3 times and with Noor 2 times. This week you hung out with Maya 4 more times and with Noor 1 more time. To total it up, you count Maya hangouts with Maya hangouts and Noor hangouts with Noor hangouts.
How it connects: Like terms are counted together: 3m + 2n + 4m + n = 7m + 3n. Hangouts with Maya do not merge with hangouts with Noor, just as m-terms do not merge with n-terms. (In algebra the letter is an unknown number, not a friend, but the grouping rule is the same.)
The Sorting Hat
In the Harry Potter stories, a talking hat sorts each new student into one of four school houses. Every student ends up in exactly one house.
How it connects: Sorting terms works the same way: each term goes into the group that matches its variable part (the x group, the x2 group, the constants group). Then, inside each group, add the coefficients, keeping their signs: in 3x + 2x − x the x group gives 3 + 2 − 1 = 4, so 4x. Terms in different groups never merge, so x and x2 stay apart.
Sesame Street’s odd-one-out game
Sesame Street has a long-running game where you look at a group of items and spot the one that does not belong with the others.
How it connects: Try it on 3x, 5x, −x and 2x2. The odd one out is 2x2, because its variable part is x2 while the others have x. The three that match are like terms: 3x + 5x − x = 7x.
Birds of a feather flock together
The proverb means that similar people tend to gather with each other.
How it connects: Like terms flock together. In 7z + 3 + 2z, the two z-terms are the birds of a feather: 7z + 2z = 9z. The constant 3 stays on its own, so the result is 9z + 3.
Apples and oranges
Someone who is “comparing apples and oranges” is comparing two things that are too different to compare sensibly.
How it connects: In the same spirit, unlike terms do not merge. If a is the price of one apple and r is the price of one orange, then 3 apples and 2 oranges cost 3a + 2r, and that is as simple as it gets. You cannot write it as 5a or 5r. (The letters stand for numbers, here prices, not for the fruit itself.)
A taco-truck receipt
A receipt shows 5 tacos and 2 sodas. Later you add 3 more tacos and return 1 soda. Tacos go with tacos: 5 + 3 = 8. Sodas go with sodas: 2 − 1 = 1.
How it connects: Let t be the price of one taco and s the price of one soda. The bill is 5t + 2s + 3t − s = 8t + s. The returned soda is −s, which means −1s, so the sodas combine as 2s − 1s = 1s, written simply as s. A letter on its own has coefficient 1. (The letters are prices, which are numbers, not the food itself.)
A season record
A team’s record lists wins and losses separately. Weekend one: 3 wins and 1 loss. Weekend two: 2 wins and 2 losses.
How it connects: Let W stand for a win and L for a loss: (3W + 1L) + (2W + 2L) = 5W + 3L. Combine wins with wins and losses with losses. The record of 5 wins and 3 losses stays as two separate counts: it does not become 8W or 8L. (Here W and L are just labels for counting. In algebra a letter stands for a number, but the rule of combining only matching kinds is the same.)
Inventory stacks
In many video games, identical items stack in one inventory slot: 12 stone blocks and 20 more stone blocks make one stack of 32. A different item, such as dirt, goes in its own slot.
How it connects: Like terms stack: 12s + 20s = 32s. The coefficient is like the stack size, and the variable part is like the item type. Even s and s2 are different variable parts, so they take different slots. (In algebra the letter stands for a number, not an item, but only matching variable parts stack.)
Memory help
SAME
Same letters And Matching Exponents. Two terms are like terms only if they pass SAME. Constants have no letters, so any two constants pass.
The like-terms rhyme
Same letters, same exponents: they go together. Add the numbers in front, keep the letters whatever the weather. Different letters or exponents: leave them alone. Side by side they stay, each on its own.
Shapes and colors
Mark each kind of term with its own shape: circle the x-terms, box the constants, underline the x2-terms. Mark the sign inside the shape too. Then add up what is inside each shape.
The name-tag party
Picture a party where every guest wears a name tag showing a variable part: x, x2, y, or nothing for constants. Each guest also holds a number card, which is the coefficient with its sign (a guest with no number shown holds a 1). Guests with the same tag sit at the same table. At each table, add up the number cards (a negative card takes away), and that total is the new coefficient. Tables never merge, so x and x2 stay at separate tables.
The invisible 1
A lone x is really 1x, and −x is −1x. When you combine 4x − x, say it out loud: four x minus one x is three x.
Loop the sign with its term
Draw a loop around each term together with the sign on its left, such as the −4u in 9 − 4u. When you slide terms next to each other, the loops move with their signs.