Like terms and simplification

You can add $3$ apples to $5$ apples and get $8$ apples. But you cannot add $3$ apples to $5$ oranges. The same logic applies to algebraic terms.

Idea

Two terms are like terms if they have the same letter(s) raised to the same power(s). Examples:

• $3x$ and $7x$ , both have $x$, so they are like.
• $4ab$ and $-2ab$ , both have $ab$, so they are like.
• $5x^2$ and $9x^2$ , both have $x^2$, so they are like.

Unlike terms have different letters or different powers:

• $3x$ and $5y$ , different letters.
• $4x$ and $4x^2$ , different powers.

To add or subtract like terms, simply add or subtract their coefficients and keep the letter part the same: $3x + 5x = (3 + 5)x = 8x, \qquad 7y - 2y = 5y.$

Unlike terms cannot be combined. You can only write them side by side as a sum: $3x + 5y$.

This idea generalises the distributive property in reverse. $3x + 5x = (3 + 5)x = 8x$ is essentially factoring out the common $x$.

When you have a long expression like $5x + 3y - 2x + 4y$, group like terms first: $5x + 3y - 2x + 4y = (5x - 2x) + (3y + 4y) = 3x + 7y.$

Worked examples

Example 1. Simplify $4a + 7a$.

Both terms have $a$: $4a + 7a = 11a$.

Example 2. Simplify $8x - 3x + 2y - 5y$.

Group: $(8x - 3x) + (2y - 5y) = 5x + (-3y) = 5x - 3y$.

Example 3. Simplify $3(x + 4) + 2x$.

Distribute: $3x + 12 + 2x$. Combine like terms: $(3x + 2x) + 12 = 5x + 12$.

Example 4. Simplify $7 + 2x + 3 - x + 4y$.

Group: $(2x - x) + 4y + (7 + 3) = x + 4y + 10$.

Try it yourself

1. Simplify $6x + 9x$.
2. Simplify $11p - 4p$.
3. Simplify $4a + 3b - a + 5b$.
4. Simplify $5(x + 2) + 3x$.
5. Simplify $2y + 7 - 5y + 3$.
6. Simplify $4x^2 + 2x^2 - x^2$.
7. Can $3x + 4y$ be simplified further? Why or why not?
8. Simplify $2(a + b) - (a - b)$.

Activity

Take a bag of buttons in two colours (say red and blue). Represent "red" as $r$ and "blue" as $b$. Pick handfuls and write expressions like $3r + 5b$. Combine bags with a friend: $(3r + 5b) + (4r + 2b) = 7r + 7b$. Watch the like terms combine.