Rectangles show why it works

Use rectangles and their areas to see why like terms can be added.

You can see why like terms add, using areas of rectangles.

Put two rectangles side by side. Both have height v; one is 4 wide, the other 3 wide. Their areas are 4v and 3v.

Together they make one big rectangle: height v, width 4 + 3 = 7, so area 7v. It is the same rectangle either way, so 4v + 3v = 7v. That is the distributive property, drawn as a picture.

Subtraction works too. Rectangle ABCD has height n and width 12. Cut off a piece EBCF of width 4. The piece left, AEFD, has width 12 − 4 = 8, so its area is 8n.

Or: whole area minus the cut piece: 12n − 4n. Same rectangle, so 12n − 4n = 8n.

Whenever you're not sure about combining like terms, draw the rectangles and check.

Adding like terms is adding the areas of rectangles that share a side — which is why 4v + 3v must be 7v.

Why like terms add.

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