Chapter 2Section 2.1

Solving Linear Equations

Isolating the VariableThe most common strategy to solve an equation is to isolate the variable. This means to get the variable on one side of the equation (and only one side) all by itself and only numbers on the other side. This is done by doing some group the following to both sides of the equation in the correct order.Add a number or its oppositeMultiply by a number or its reciprocalAdd an expression or its oppositeCombine like termsDistribute numbers into grouping

symbols to eliminate the grouping.

Solve for x,

Add 8 to both side of the equation,

Solve for x,

Multiply both sides by ,

Order of OperationsWhen solving for a variable the order in which you isolate the variable is the opposite of the order of operations.1) Addition and

Subtraction2) Multiplication, Division

and Negative signs3) Exponents4) Grouping (Although the

first step will be to eliminate the grouping)

Solve for y,

Worry about addition first. Add to both sides,

Solve for a,

Change to addition. Add constants (numbers).

Take negative of both sides,

Combining Like TermsTerms that are like or alike are expressions that have the exact same variable(s) with exactly the same exponents on the variable(s). If a number is in front of the variable(s) we call that the coefficient. Like terms being added or subtracted can be consolidated by adding or subtracting their coefficients as the sign on the coeffients would suggest.

Like Terms:

Non-like Terms:

Solve,

Solve, (Usually do not show were you add to get zero or multiply to get 1.)

Removing Grouping SymbolsIn order to remove grouping symbols that have a number or a negative being multiplied outside the symbols. You need to distribute the number or negative by multiplying each term inside the symbols by the number to and removing the grouping in order to combine like terms.Distribute a 5,

Solve,

Clearing FractionsIf an equation has fractions in it do not try to combine terms with fractions. Get rid of all the fractions by multiplying by a big enough number that will cancel out all of the fractions.

Solve,

Solve,

Variables Cancel OutSometimes when you try to solve an equation all of the variables will cancel out and leave just numbers. If what is left are two equal numbers the equation is called an identity and all numbers are a solution. If what is left are two unequal numbers the equation is called a contradiction and there is no solution.

Solve,

Solve,

This is an identity every number will be a solution.

This is a contra-diction no number is a solution.