Algebra Overview part 4

Simplfying by multiplication
When solving for a variable, we want to get a solution like x = 3 or z = 2001. When a variable is divided by some number, we can use multiplication on both sides to solve for the variable. Example:

Solve for x in the equation x ÷ 12 = 5.

Since the x on the left side is being divided by 12, the equation is the same as x × 1/12 = 5. Multiplying both sides by 12 will cancel the 1/12 on the left side:

x × 1/12 × 12 = 5 × 12 ==>

x × 1 = 60 ==>

x = 60.

Simplifying by division
When solving for a variable, we want to get a solution like x = 3 or z = 2001. When a variable is multiplied by some number, we can use division on both sides to solve for the variable. Example:

Solve for x in the equation 7x = 133. Since the x on the left side is being multiplied by 7, we can divide both sides by 7 to solve for x:

7x ÷ 7 = 133 ÷ 7 ==>

(7x)/7 = 133 ÷ 7 ==>

x/1 = 19 ==>

x = 19.

Note that dividing by 7 is the same as multiplying both sides by 1/7.

Word problems as equations
When converting word problems to equations, certain "key" words tell you what kind of operations to use: addition, multiplication, subtraction, and division. The table below shows some common phrases and the operation to use. Word Operation Example As an equation sum addition The sum of my age and 10 equals 27. y 10 = 27 difference subtraction The difference between my age and my younger sister's age, who is 11 years old, is 5 years. y - 11 = 5 product multiplication The product of my age and 14 is 168. y × 14 = 168 times multiplication Three times my age is 60. 3 × y = 60 less than subtraction Seven less than my age equals 32. y - 7 = 32 total addition The total of my pocket change and 20 dollars is $22.43. y 20 = 22.43 more than addition Eleven more than my age equals 43. 11 y = 43

Sequences
A sequence is a list of items. We can specify any item in the list by its place in the list: first, second, third, fourth, and so on. Many useful lists have patterns so we know what items occur in each place in the list. There are 2 kinds of sequences. A finite sequence is a list made up of a finite number of items. An infinite sequence is a list that continues without end. Examples:

The following are examples of finite sequences.

ADVERTISEMENT:

The sequence 1, 3, 5, 7, 9, 11, 13, 15, 17, 19 is the sequence of the first 10 odd numbers.

The sequence a, e, i, o, u, is the sequence of vowels in the alphabet.

The sequence m, m, m, m, m, m is the sequence of 6 m's.

The sequence 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0 is the sequence of 12 alternating 1's and 0's.

The sequence 1, 2, 3, 4, ..., 9998, 9999, 10000 is the sequence of the first ten thousand integers.

The sequence 0, 1, 4, 9, 16, 25, 36, 49 is the sequence of the squares of the first 8 whole numbers.

Examples:

The following are examples of infinite sequences.

The sequence 2, 4, 6, 8, 10, 12, 14, 16, ... is the sequence of even whole numbers. The 100th place in this sequence is the number 200.

The sequence a, b, c, a, b, c, a, b, c, a, b, ... is the sequence of the letters a, b, c, repeating in this pattern forever.

The 100th place in this sequence is the letter a. The 300th place in this sequence is the letter c.

The sequence -1, 2, -3, 4, -5, 6, -7, 8, -9, ... is the sequence of integers with alternating signs. The 10th place in this sequence is 10. The 100th place in this sequence is 100. The 101st place in this sequence is -101.

The sequence 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, ... is a sequence of 1's separated by 1 zero, then 2 zeros, then 3 zeros, and so on. The 100th place in this sequence is a 0. The 105th place in this sequence is a 1.

The sequence 1, 3, 6, 10, 15, 21, 28, 36, 45, ... is the sequence of places the 1 occurs in the sequence of 1's and 0's above! If this sequence seems strange, note the difference between pairs of numbers next to one another:

3 - 1 = 2

6 - 3 = 3

10 - 6 = 4

15 - 10 = 5

21 - 15 = 6

28 - 21 = 7

Checking these differences makes the pattern clearer.

1, 1, 1, 1, 1, 1, ... is the sequence where every item in the list is the number 1.

1, 2, 3, 4, 5, 6, 7, ... is the sequence of counting numbers. Each item in the list is its place number in the list.

a, b, a, b, a, b, a, b, ... is the sequence of alternating letters a and b. The a's occur in odd-numbered places, and the b's occur in the even-numbered places.

1/1, 1/2, 1/3, 1/4, 1/5, 1/6, 1/7, ... is the sequence of reciprocals of the whole numbers.

1, 4, 9, 16, 25, 36, 49, 64, 81, ... is the sequence of squares of the whole numbers.

a, e, i, o, u, a, e, i, o, u, a, e, ... is the repeating sequence of vowels in the alphabet.

4, 7, 10, 13, 16, 19, 22, 25, ... is the sequence of numbers beginning with the number 4, and each number in the list is 3 more than the number before it.