Variables
A variable is a symbol that represents a
number. Usually we use letters such as x,y,z,s,etc., for variables.
For example, we might say that s stands for the side-length of a square.
The perimeter of the square is given by 4 × s. The area of the square is
given by s2.
When working with variables, it is useful
to use a letter that will remind you of what the variable stands for: let n
be the number of people in a movie theater; let t be the time
it takes to travel somewhere; let d be the distance from my house
to the park.
Expressions
An expression is a mathematical statement using
numbers, variables, or both.
Example:
The following are examples of expressions:
6
x
3 + 7
6 × y + 5
(9-5) × 14
Example:
Roland weighs 70 kilograms, and Mark weighs
k kilograms. Write an expression for their combined weight. The combined
weight in kilograms of these two people is the sum of their weights, which is
70 + k.
Example:
A car travels down the freeway at 55
kilometers per hour. Write an expression for the distance the car will have
traveled after h hours. Distance equals rate times time, so the distance
traveled is equal to 55 × h..
Example:
Evaluate the expression 4z+ 16 when z
= 15.
We replace each occurrence of z with
the number 15, and simplify using the usual rules: parentheses first, then
exponents, multiplication and division, then addition and subtraction.
4z + 16 becomes
(4 × 15) + 16 = 76
Equations
An equation is a statement that two numbers
or expressions are equal. Equations are useful for relating variables and
numbers. Many word problems can easily be written down as equations.
Example:
The following are examples of equations:
6 = 6
21 = 6 + 15
x = 7
7 = x
t + 3 = 8
3 (n +16) = 100
w + 4 = 16 - w
y - 1 - 6 - 9.3 = 34
3 (d + 4) - 11 = 4
Example:
Translate the following word problem into
an equation:
My age in years y plus 60 is equal
to four times my age, minus 12.
The first expression stands for "my
age in years plus 60", which is y + 60.
This is equal to the second expression for
"four times my age, minus 12”, which is 4y – 12.
Setting these two expressions equal to one
another gives us the equation:
y + 60 = 4y - 12
Solution of an
Equation
When an equation has a variable, the
solution to the equation is the number that makes the equation true when we
replace the variable with its value.
Example:
We say y = 3 is a solution to the
equation 4y + 7 = 19, since replacing y with 3 gives us
4 × 3 + 7 = 19 ==>
12 + 7 = 19 ==>
19 = 19 which is true.
Examples:
x = 100 is a solution to the equation (x /2) - 40 = 10
z = 16 is a solution to the equation 5 × (z - 6) = 50
Counterexample:
y = 10 is NOT a solution to the equation 4y + 7 = 19. When we
replace each y with 10, we get
4 × 10 + 7 = 19 ==>
40 + 7 = 19 ==>
47 = 19 not true!
Counterexamples:
x = 600 is NOT a solution to the equation x ÷ 6 - 40 = 10
z = 6 is NOT a solution to the equation 5 × (z - 6) = 30
Simplifying Equations
To find a solution for an equation, we can
use the basic rules of simplifying equations. These are as follows:
1) You may evaluate any parentheses,
exponents, multiplications, divisions, additions, and subtractions in the usual
order of operations. When evaluating expressions, be careful to use the
associative and distributive properties properly.
6) You may combine like terms. This means
adding or subtracting variables of the same kind. The expression 6x + 4x
simplifies to 10x. The expression 13 - 7 + 3 simplifies to 9.
3) You may add any value to both sides of
the equation.
4) You may subtract any value from both
sides of the equation. This is best done by adding a negative value to each
side of the equation.
5) You may multiply both sides of the
equation by any number except 0.
6) You may divide both sides of the equation
by any number except 0.
Hint: Since subtracting any number is the
same as adding its negative, it can be helpful to replace subtractions with
additions of a negative number.
Example:
This problem illustrates grouping like
terms and dealing with subtraction in an equation.
Solve x - 12 + 20 = 37.
x + 8 = 37.
Now we may subtract 8 from each side of the
equation, (we will actually add a -8 to each side).
x + 0 = 29
x = 29
Example:
This problem illustrates the proper use of
the distributive property.
Solve 2 × (x + 1 + 4) = 20.
2(x + 5) = 20
The equation now becomes
2x + 10 = 20.
Subtracting a 10 (adding a -10) to each
side gives us
2x = 10.
Since the x is multiplied by 2, we
divide both sides by 2 to solve for x:
x = 5.
We can check this solution in the original
equation:
2 × (5 + 1 + 4) = 20 ==>
2 × 10 = 20 ==>
20 = 20 so our solution is correct.
Combining like terms
One of the most common ways to simplify an
expression is to combine like terms. Numeric terms may be combined, and any
terms with the same variable part may be combined.
Example:
Simplify the expression 4+ 2x + 8 -
3x – 5
= (4 + 8 – 5) + (2x – 3x)
= 7 + (-x) = 7 - x
Examples:
For the equation 3x + 4 = 12, we can
isolate the variable term on the left by subtracting a 4 from both sides:
3x + 4 - 4 = 12 - 4 ==>
3x = 8
x = 8/3
Simplfying by
multiplication
When a variable is divided by some number,
we can multiply both sides to solve for the variable.
Example:
Solve for x in the equation x /12 = 5.
Multiplying both sides by 12 will cancel
the 12 in the denominator:
x/12 × 12 = 5 × 12 ==>
x = 60.
Simplifying by
division
When solving for a variable, we want to get
a solution like x = 3 or z = -6. 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 = 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 20 equals 52. |
y + 20 = 52 |
|
difference |
subtraction |
The difference between my age and my
younger brother's age, who is 9 years old, is 3 years. |
y - 9 = 3 |
|
product |
multiplication |
The product of my age and 14 is 280. |
14y = 280 |
|
times |
multiplication |
Three times my age is 20. |
3y = 20 |
|
less than |
subtraction |
Seven less than my age equals 21. |
y - 7 = 21 |
|
total |
addition |
The total of my pocket change and 10 dollars
is $10.67. |
y + 10 = 10.67 |
|
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 sequences 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.
The sequence 1, 3, 5, 7, 9, 11, 13, 15, 17,
19 is the sequence of the first 10 odd numbers.
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.
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.
1/1, 1/2, 1/3, 1/4, 1/5, 1/6, 1/7, ... is
the sequence of reciprocals of the whole numbers.