Simplifying Algebraic Expressions

Why simplifying matters

I'm thinking of a number. Two of it, plus three more of it, makes 25. What's the number?

The two x-terms on the left are the same kind of thing: two of x, then three more of x. Five x's in total. Before solving, rewrite the left side that way:

Now it's a one-step problem. Rewriting an expression in a shorter shape that names the same value is called simplifying, and it's what this lesson is about.

Two errors catch most of Grade 7 here, and both get names today: gluing unlike terms (writing 5a + 3b = 8ab, as if + were a multiplication) and partial distribution (writing 3(x + 2) = 3x + 2, where the 3 quits after the first term). Watch for both — the workbench below is rigged to catch them.

How combining like terms works

The picture for 2x + 3x = 5x is a rectangle with height x and width 5. Two x-tiles laid next to three x-tiles cover the same area as five x-tiles in a row.

Drag the split. The numbers on either side change, the total doesn't. 2x + 3x, 1x + 4x, and 5x all name the same area.

The rule behind this is the distributive property:

Read left-to-right, it expands a bracket: 3(x + 2) becomes 3x + 6. Read right-to-left, it does the move you just did:

Try it on the workbench

Each term becomes a tile. The rectangle's shape encodes the letter; its colour encodes the sign. Orange is positive, prairie is negative. Like tiles snap into the same row — but the button won't snap them for free: call the simplified form first, and the tiles only move once your call is right. A wrong call gets told exactly which trap it fell into.

The first scene starts with 4x + 7 + 2x − 3. Six x-tiles total in one row, four units left after three negative units cancel three of the seven positive units. Result: 6x + 4.

The second scene has two letters. The a-tiles and b-tiles never share a row: different shape, different family. 5a + 3b − 2a + 4b − 1 simplifies to 3a + 7b − 1.

The third scene shows that decimal coefficients work the same way. A half-tile is literally half-width; two half-tiles make one full x-tile. 0.5x + 1.25 + 1.5x − 0.25 simplifies to 2x + 1.

Distribution as duplication

Wrap a set of tiles in a multiplier frame and pull it outward. The contents physically duplicate. That is the distributive property — and here too, the Distribute button demands your call before it moves. The wrong options are the partial-distribute and sign-flip errors, so picking carelessly walks you straight into the trap with the lights on.

4(2x − 5) becomes four copies of (2x − 5): eight x-tiles and twenty negative units. Result: 8x − 20. Notice the 4 lands on both the 2x and the −5. Distributing onto only the first term and writing 8x − 5 is the most common error in this lesson.

The third scene flips signs. A −2 outside the parentheses doesn't just multiply by 2. It flips the colour of every inner tile. −2(4 − 3y) becomes −8 + 6y. Both terms inside change sign, not just one.

Where it shows up in real life

A school fundraiser sells two items: granola bars at $x each and juice boxes at $y each. One Grade 7 class sells 5 bars and 3 juice boxes. Another class sells 2 bars and 4 juice boxes. Total revenue is 5x + 3y + 2x + 4y, which simplifies to 7x + 7y: seven of each, same as if the two classes had pooled their orders before selling.

A taxi in Calgary charges a flat $3.95 plus $2.10 per kilometre. For a ride of d kilometres the fare is 3.95 + 2.10d. Two riders splitting one taxi pay (3.95 + 2.10d) ÷ 2, which the distributive property rewrites as 1.975 + 1.05d: half the flat charge plus half the per-kilometre rate.

The point isn't the specific dollar amount. It's that simplifying gives a shorter expression that names the same value, with the same arithmetic baked in.

Is the answer reasonable?

Substitute a test value. If two expressions are claimed equivalent, they must agree at any number you pick.

Pick x = 4. Then 3(x + 2) = 3(6) = 18, and 3x + 6 = 12 + 6 = 18. Match. Try x = 4 against the wrong rewrite: 3(x + 2) = 18 versus 3x + 2 = 14. Mismatch, so they aren't equivalent. The test value rules out the partial-distribution error directly.

One match doesn't prove equivalence (two different expressions can agree by accident at a single input), but one mismatch disproves it. Substitution is a fast, cheap way to flag a wrong simplify.

True or false?

Five rounds. Each shows two expressions joined by =. Read = as "is the same as," not "the answer is on the right." The fourth round is the trap, so be careful.

Misconception probe

Worksheet

These aren't graded. Get them right, get them wrong. The goal is to feel out where the idea works and where the like-terms rule kicks in.

Going further

Next: solving equations. Simplifying lets you rewrite each side without changing what it means; solving uses the same idea but acts on both sides at once. The rule that the workbench enforces here ("like tiles can combine") becomes the rule that the next lesson enforces on a balance ("the same move on both sides keeps the scale level").

The distributive property reappears in fractions (a · b/c = ab/c), in arithmetic with money (split a flat fee plus a per-unit rate), and in geometry (area of a composite rectangle).