Download Solving Equations

Variables
What is a variable? It is a box, waiting for a value. Sometimes the value is already inside the box, and you
have to figure out what the value is. Other times, the box is empty, and you get to pick the value to put
inside. More about that later. First... Remember when you were in elementary school, and you were
learning your addition? The teacher would hand you worksheets that said things like:
+ 2 = 5 ; fill in the box.
Variables are the same thing. Now we say:
x + 2 = 5 ; solve for x.
Why did we switch from boxes to letters? Because letters are better. Boxes come in only a few shapes, but
letters come in many varieties, and letters can stand for something.
A variable is a letter that stands for a number whose value we don’t know yet. It’s a “nickname” for
an unknown number.
In the first case, "x + 2 = 5", x can only be 3. The statement is not true for any other value. That is to
say, the value of x is "fixed"; we just have to figure out what it is. The value of the variable that makes
the statement true is called the solution.
What is the solution to
n + 6 = 9?
Now that we have variables, what do we do with them? Go back in your mind again to elementary school.
Your teacher would have you add "2 apples plus 6 apples is 8 apples". The same rules apply to
variables: "2 boxes plus 6 boxes is 8 boxes", or, using variables, "2x + 6x = 8x". "A box and another
box is two boxes", or "x + x = 2x". Think of x as 1x, so x + x = 1x + 1x = 2x.
But note: "2 apples plus 6 oranges" is just 2 apples and 6 oranges; they might make a nice fruit salad, but
they're not 8 of anything. In the same way, "2x + 6y" is just 2x + 6y; you can't combine the two
variables into one.
When multiplying, we use exponents. For instance, (5)(5) = 52. Of course, we can simplify this as 52 = 25.
Similarly, (x)(x) = x2. But, until we know what value to put in for x, we cannot simplify this.
Don't confuse multiplication and addition: (x)(x) does not equal 2x, just as (5)(5) does not equal (2)(5), so
(x)(x) equals x2. (Note the technique I just used: If you're not sure what to do with the variables, put in
numbers, where you know what to do. Then, whatever you did with the numbers, do that with the
variables.)
Expressions and Equations
Expressions (contain no “=” sign) :
An expression is one or numbers having some mathematical operations done on them.
Numerical Expressions:
3+5
3(4)
6/2
5-1
4
Expressions can be evaluated or simplified: 3 + 5 can be simplified to 8
This just means, “Whatever you see, do”
Algebraic Expressions:
x+5
3x  in Algebra, it is implied that 3x means 3 times x. The “proper” way to write the product of a number
and a variable is to always write the number to the left of the variable. x times 5 = 5x
When numbers are multiplied by variables, they are given a special name, “coefficient”. 5 is the coefficient
of 5x.
Algebraic expressions can only be simplified if they have “like terms,” which are terms with the same
variable and same exponent.
x+5+2
Only 5 + 2 can be simplified. x + 5 + 2 becomes x + 7
3x + x + 2
can be simplified to 4x + 2
2
x +x
cannot be simplified because they don’t have the same exponents and are therefore not like terms.
Equations:
An equation is two expressions set equal to one another.
Numerical Equations:
4=5-1
8=3+5
2 + 2 = 5 -1
12 = 3(4)
6+2=3+5
3 = 6/2
Algebraic Equations:
x = 3(4)
y–3=4
Equations can be true or false.
True: 3 = 2+1
False: 2+3 = 2(3)
Algebraic Equations can be true or false depending on the value of the variable.
y – 3 = 4 is true if y = 7 and false if y equals anything else.
Some algebraic equations are false no matter what: y = y + 2 can never be true no matter what y is.
Algebraic Equations can be solved. A solution to an equation is the value of the variable that makes the
equation true.
Two equations are equivalent equations when they have the same solution.
x + 1= 4 is equivalent to x + 2 = 5
In both cases, x must be 3 for the equation to be true.
How do you solve equations?
Get x by itself one side of the equation (isolate x) so you are left with x = ___. The right side of the
equation will just have a numerical expression that can be simplified.
Solving Equations
Subtraction Property of Equality –
Subtracting the same thing from both sides of an equation does not change the solution.
x+2 = 5
Subtract 2 from both sides:
x+2–2=5–2 =3
This new equation will have the same solution as x + 2 = 5.
x+0=3
x=3
Hey! We just “isolated” x!
Check your answer by substituting x=3 in the original equation.
3 + 2 = 5? Yes!
Soooo…., if you have an equation with a number added to a variable on one side,
subtract that number from both sides to get the variable by itself, or to “isolate the variable."
But that example was too easy. Try this one:
x + 15 = 67
Addition Property of Equality –
Adding the same thing to both sides of an equation does not change the solution.
x – 17 = 34
Add 17 to both sides.
x – 17 + 17 = 34 + 17
This new equation will have the same solution as x – 17 = 34.
x + 0 = 51
x = 51
Check your answer by substituting x = 51 in the original equation.
51 – 17 = 34? Yes!
Soooo…., if you have an equation with a number subtracted from a variable on one side,
add that number to both sides to get the variable by itself.
If an equation has more than one operation in the expressions on both sides, simplify the expressions
as much as possible before isolating the variable. If the variable is already isolated on one side but
other side has an expression, just simplify that expression and VOILA! You’ve got a solution!
Solve the equation
15 + 6 + n = 30
This expression can be simplified to 21
21 + n = 30
21 – 21 + n = 30 – 21
n=9
Check your solution:
Does 21 + 9 = 30? Yes!
Solve the equation
17 + 26 + 58 = n
This expression can be
simplified to 100
100 = n
Division Property of Equality –
Dividing the same thing on both sides of an equation does not change the solution.
5x = 15
5x = 15
5
5
x 3
Check : 5(3) 15 ? Yes!
If you have an equation with a number multiplied by a variable, divide both sides of the equation by
that number to get the variable by itself.
Examples of English words for multiplication: multiplied by 15 (15x), increased by a factor of six (6x),
doubled (2x) , tripled (3x), quadrupled (4x), increased fivefold (5x).
Multiplication Property of Equality –
Multiplying the same thing on both sides of an equation does not change the solution.
(5)
x
20
5
x
20(5)
5
x 100
Check :
100
5
20 ? Yes!
If your equation has a variable divided by a number, multiply both sides by that number to isolate the
variable.
Examples of English words for division: halved (x/2), a third of (x/3), split in four (x/4)
x
13
3
x
3
3(13)
3
x 39
Check :
39
13? Yes!
3
Notice that x divided by 3 is written as a fraction, x/3.
When multiplying a whole number by a fraction, think of the whole
number as a fraction with denominator 1. For example 3 = 3/1, so
3
x
3
3
1
x
3
x
1
x