3

(1/2)(k ÷ 6)

2
1
Guided Practice

Suppose a rancher has 3 sheep and another rancher has 4 sheep. How many sheep are there in all? Seven, of course! The same kind of thinking can be applied to algebra: 3s + 4s = 7s! The terms 3s and 4s are called like terms. When you combine like terms you add or subtract them.

What if a rancher has 3 pigs, and another rancher has 4 chickens? It's hard to even ask a question using addition or subtraction: how many pig-chickens in all? It doesn't make sense. In the same way, 3p + 4c cannot be simplified by adding. The terms 3p and 4c are unlike terms.

If the terms are like terms, add or subtract them. If they are unlike terms, write unlike.

5q + 2r =

6c + 7c =

8g – 5g =

The digits after the decimal point are called the decimal digits. If the decimal digits of a number stop at some point, the decimal is called a terminating decimal. If some or all of the decimal digits repeat forever, the number is called a repeating decimal.

The decimal 0.3256... can also be written as 0.3256. All terminating and repeating decimals are rational numbers.

A rational number is a number that can be written as the ratio of two .

The decimal 0.12 is a rational number, because it can be written as / .

The number 67 is a rational number, because it can be written as / .

When a digit has a bar over it (for example, 5), that means the digit is repeated forever.

The number 0.5 is a rational number, because it can be written as / .

The number 1.5/3 is a rational number, because it can be written as / .

Write the rational number as a ratio of two integers.

The number 4 is a rational number, because it can be written as / .

Write the rational number as a ratio of two integers.

The number 1.25 is a rational number, because it can be written as / .

Write the rational number as a ratio of two integers.

The number 0.3 is a rational number, because it can be written as / .

Write the rational number as a ratio of two integers.

The number –0.1 is a rational number, because it can be written as / .

Write the rational number as a ratio of two integers.

The number –100 is a rational number, because it can be written as / .

Write the rational number as a ratio of two integers.

The number 0.75/0.25 is a rational number. It can be written as / .

Use long division to decide whether the rational number is a terminating or repeating decimal. Write terminating or repeating.

The number 2/3 is a decimal.

Use long division to decide whether the rational number is a terminating or repeating decimal. Write terminating or repeating.

The number 5/8 is a decimal.

Use long division to decide whether the rational number is a terminating or repeating decimal. Write terminating or repeating.

The number 5/6 is a decimal.

Use long division to decide whether the rational number is a terminating or repeating decimal. Write terminating or repeating.

The number –8/3 is a decimal.

When you are asked to multiply the number of letters in your first name by 2 and then by 3 separately, and then add the two results, you are combining like terms. If n stands for the number of letters in your first name, then 2n + 3n = 5n.

The terms 2n and 3n are like terms because they have the same variable expression (n) multiplied by a number. So, you can combine them into 5n by adding them.

Here is another example of combining like terms:

4(x – 1) + 4(x – 1) = 8(x – 1)

Once you have combined like terms to get 5n, the next steps in the magic trick are all algebra too. We add 2, then multiply the result by 2. We can use the Distributive Property to show the result.

Then, subtract 4 from 10n + 4 to get 10n. And, finally, divide 10n by the number of letters in your first name, n, to get 10!

The magic trick always results in , no matter what positive number you start with.

Write the sum of the like terms.

3n + 5n =

Write the sum of the like terms.

6(b + 5) + 17(b + 5) =
( + )

Write the sum of the like terms.

9(r + 8) + 7(8 + r) =
( + )

Write the sum of the like terms.

10y + y =

Write the sum of the like terms.

5(3q) + 5(3q) = ()

We can add negatives to combine like terms too. This means that we can also subtract like terms.

Write the sum of the like terms.

20(h – 8) + –5(h – 8) =
()

Write the difference of the like terms.

18(j + 4) – 12(4 + j) =
( + )

Write the sum or difference of the like terms.

25n – n =

Write the sum or difference of the like terms.

g(6 + 1) + –15g =

Use the Distributive Property to rewrite the product as a sum.

5(3n + 1) = +

Use the Distributive Property to rewrite the sum as a product.

8n + 4 = 2( + )

Use the Distributive Property to rewrite the difference as a product.

3 – 24p = 3()

Like terms are not just terms like 3g and 4g or 2x and x. Like terms can also be expressions such as 4(3 + m) and 2(3 + m). What makes them like terms is that one of the factors in each expression is the same.

You can also have like terms with fraction coefficients. Try adding and subtracting these kinds of like terms.

Write the sum of the like terms. (Write the fraction in simplest form.)

(1/2)(k ÷ 6) + (k ÷ 6)(1/4) =
( ÷ )

Write the sum or difference of the like terms. (Write the fraction in simplest form.)

(7/8)(3t – 1) – (1/8)(3t – 1) =
()

Use the Distributive Property to rewrite the product as a difference.

(1/2)(10n – 6) =

Use the Distributive Property to rewrite the product as a sum.

1.5(k + 2) = +

=

You can enter numbers, lowercase variables, or addition expressions into each side of the interactive balance on the left—expressions like 6, p, r + 2, or 3x + 1. Then press Enter to show the expression on the balance. No subtraction expressions or parentheses, though.

Coefficients (the numbers that are multiplied to variables; like the 6 in 6x) can be whole numbers from 0 to 6, and constants can be whole numbers from 0 to 6.

Try this. What value for the variable x makes the equation 3x + 1 = 6 + 4 balance? Highlight an x on the balance and type a number to replace it. Then see if you're right.

x =

Use the balance to determine the solution of the equation b + 4 = 6 + 2.

b =

Use the balance to determine the solution of the equation 4b + 3 = 6 + 5.

b =

Use the balance to determine the solution of the equation 5 + 4 = 2h + 1.

h =

Use the balance to determine the solution of the equation 6 + 2 = 5w + 3.

w =

Use the balance to determine the solution of the equation 4k = 6 + 6.

k =