Solving equations

Definitions

Expression

= any form of showing a mathematical value

  • ex. the number 2 may be "expressed" as either "2" or "1+1"
  • more complex "expressions" involve variables, such as "2y -5 = 10"
    • here, the value (expression) "10" can also be "expressed" as "2y - 5"

Equation

= a statement that uses an equal sign (=)

  • which means that the expressions on both side of the equal sign have the same value

Inverse Operation

= a method for isolating variables by adding or multiplying a value to both sides of an equation

  • the "inverse operation" reduces the value of the property on the side of the variable to 1 or 0
  • that way the variable becomes "isolated" on one side of the equation
  • ex.:
  • the "inverse operation" adds -3 to both sides of the equation:
  • which leaves us with

Property

  • = the rule that is applied to numbers in an equation
  • the property applied must be the same for both sides of the equation!
  • properties include
    • Addition property (or subtraction)
    • Multiplication property (or division)

Isolating the variable

  • Equations are solved by "isolating the variable"
    • which means "expressing" the unknown value by itself on one side of an equation
      • ex. to solve, "4 + x = 6" , we want to "isolate" x, so that we have "x = ___"

Properties of Equality

  • property = a rule
  • equality = that both sides of the equation (equal sign) have the same value

Variable

  • an unknown value represented, usually represented by the letter

How to solve an equation

Using Addition Property

  • when solving for when is added or subtracted to/from another number
    • we "isolate " by using the "Inverse Operation" to remove the number from the side with the variable,
    • note that
      • addition is adding a positive number:
        • where means "positive 3"
      • subtraction is adding a negative number:
        • where means "negative 3"
  • examples:
Properties of Equality
Addition Property
Equation
Inverse Operation add to both sides

(i.e. subtract 4)

add to both sides

(i.e. add 3)

Solution
  • another way to look at the Inverse Operation, using the same equations is:
Properties of Equality
Addition Property
Equation
Inverse Operation add to both sides

(i.e. subtract 4)

add to both sides

(i.e. add 3)

simplify simplify
Solution

Using Multiplication Property

Properties of Equality
Multiplication Property
Equation
Inverse Operation multiply both sides by 6

(isolates x by making the expression which is equal to )

divide both sides by 6

(isolate by making the expression which is equal to )

cancel

(because )

cancel

(because )

(note: is the same as )

Solution

Cross-multiplying to solve for when is a denominator (bottom of a fraction)

numerator

  • numerator is at the top of the fraction

  • the numerator represents the number being divided by another number
  • i.e., = the same as saying,

denominator

  • the denominator is at the bottom of the fraction

  • the denominator represents the number dividing into the other number
  • .e., = the same as saying,

How to solve for when is a denominator: "Cross-multiplication"

Using Cross-Multipliclation
is the same as
using cross-multiplication, we can move the variable to the top of the fraction (numerator)
is the same as

(using cross-multiplication)