Combining Like Terms

Using the language of algebra

Combining Like Terms

Have you ever heard the phrase “You are trying to compare apples to oranges”.

Explain what you think this phrase means.

Apples and oranges cannot be compared because they are unlike objects.

Term – the parts of an expression that are added or subtracted.

(x + 2) (2x – 4) Like terms – 2 or more terms that have the same variable raised to the same power. (in the expression 3a + 5b + 12a, 3a and 12a are like terms.) To simplify an expression – perform all possible operations, including combining like terms.

x Add or Multiply?

x + x

1x + 1x = 2x

x x x

x

x + y

Add or Multiply?

1x + 1y = x + y

y x y

A procedure frequently used in algebra is the process of combining like terms. This is a way to “clean-up” an equation and make it easier to solve. For example, in the algebraic expression 4x + 3 + 7y , there are three terms: 4x, 3, and 7y .

Remember the 4 and 7 are coefficients.

Let’s say we are given the equation below. It looks very complicated, but if we look carefully, everything is either a constant (number), or the variable x with a coefficient (4x).

Remember, a coefficient is the number by which a variable is being multiplied (the 4 in 4x is the coefficient)

The “like terms” in the equation are ones that have the same variable . All constants are like terms as well.

This means 15, 10, 6, and -2 are all like terms, and the other is 4x, -3x, 5x, and 3x. To combine them is pretty easy, you just add them together and make sure they are all on the same side of the equation.

Since the 15 and 10 are both constants we combine them to get 25. The 4x and -3x each have the same variable (x), so we can add them to get 1x. Doing the same on the other side we arrive at 25 + 1x = 4 + 8x. The process is still not finished.

There are still some like terms, but they are on opposite sides of the equal sign. Since we can do the same thing to both sides we just subtract 4 from each side and subtract 1x from each side. What remains is 21 = 7x.

Now it’s just a simple process of dividing by seven on each side and we arrive at our answer of x = 3.

Combining like terms enables you to take that huge mess of an equation and make it something much more obvious to solve.

Combine the following: 1) 14a – 5a 2) 7x – 3x 3) 12g + 7g 4) 7y + 8 – 3y – 1 + y 5) 5t + 7p – 3p -2t

Simplify Algebraic Expressions by combining like terms.

6 Simplify: 6(n + 5) – 2n = (n) + 6 (5) – 2n = Distributive Property 6n + 30 – 2n = 6n and 2n are like terms 4n + 30 Combine coefficients 6 – 2 = 4 Remember that a term like “x” has a coefficient of 1, so terms such as x, n, or y can be written as 1x, 1n, or 1y.

Example: 2a + 5b + 5 – a + 3

How many terms are in this expression?

What are the like terms?

Simplify by combining like terms.

a + 5b + 8

Example: 2 + 8(3y + 5) - y

What would be the first step in simplifying the 8 expression?

Use the Distributive Property to simplify 8(3y + 5), (3y) + 8 (5), 24y + 40 Combine like terms.

2 + 24y + 40 –y 42 + 23y

Math Humor

Teacher: Sam, please evaluate x + 4 if x = 7.

Sam: Is this a trick question? Yesterday you told us that x = 2!