Solving Equation Notes

An Intro to Equations
An equation is a mathematical statement that contains an = sign.
An equation is true when the expressions on either side of the = sign have equal value.
17 – 4 = 11 + 2
2x + 3x = 5x
42 + 4 = 10 + 10
An equation is false when the expressions on either side of the = sign are NOT equal.
22 – 10 = 7 + 7
x + 4x = 6x
42 + 10 = 25 + 5
An equation is an “open sentence” if it contains an unknown variable, and therefore could be
true or false, depending on the value of the variable.
Examples: 4x + x = 5x is true.
14 – 10 = 14 is false.
2x – 14 = 54 is an open sentence, because it includes a variable, whose
value is unknown.
Remember – a solution to an equation containing a variable, is the value of the variable that
makes the equation true.
Many equations can be solved using mental math. For example:
Solve for x, given x + 14 = 36
and, solve for z, given 5 + 3z = 20
When equations involve more than a couple “mental” steps, it is important to understand the
steps to follow when solving an equation for a given variable.
We need to know:
the same quantity can be added to both sides of an equation without changing the
solutions to the equation; and
every term on both sides of an equation can be multiplied or divided by the same
quantity without changing the solutions to the equation.
To solve an equation, we reverse the normal order of operations after eliminating
parentheses, by undoing addition or subtraction first (of what’s being “done to” the variable),
then we undo multiplication or division (of the coefficient of the variable).
When solving equations – ALL OTHER RULES APPLY. (For example: finding common
denominators when combining fractions; converting mixed numbers to improper fractions
when multiplying/dividing fractions; decimals must be aligned when adding/subtracting OR
eliminated, if preferred.)
Follow these steps, when solving equations (ALWAYS start with asking yourself,
“are there any parentheses that need to be eliminated?” and when a step is not
needed, simply move on to the next one):
1. Eliminate parentheses, if there are any (remember we do this either through
using the Distributive Property or by “taking the opposite of” if there is a
negative sign directly outside the parentheses).
2. Combine like terms on each side of the equation, if there are any.
3. Eliminate the variable on one side of the equation, if it’s on both sides of the
equal sign (we do this to get the variable alone on the other side of the
4. Move the constant (#’s) to the opposite side of the variable in the equation,
to isolate the variable on one side of the equal sign.
5. Eliminate the coefficient of the variable either through inverse operations or
by multiplication of the reciprocal of the coefficient.
6. Reduce, if possible.