Algebra

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.:
  • $x+3=8$
• the "inverse operation" adds -3 to both sides of the equation:
  • $x+3-3=8-3$

• which leaves us with
  • $x+0=5$
  • $x=5$

Operation

= a process to change a value

• addition, subtraction, multiplication and division are the fundamental "operations" of math

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 $x$

How to solve an equation with a single variable

Using Addition Property

• when solving for $x$ when $x$ is added or subtracted to/from another number
  • we "isolate $x$" by using the "Inverse Operation" to remove the number from the side with the variable, $x$
  • note that
    • addition is adding a positive number: $5+(+3)=8$
      • where $(+3)$ means "positive 3"
    • subtraction is adding a negative number: $5+(-3)=2$
      • where $(-3)$ means "negative 3"
• examples:

Properties of Equality

		Addition Property
Equation		$x+4=6$		$x-3=4$
Inverse Operation	add $-4$ to both sides (i.e. subtract 4)	$x+4+(-4)=6+(-4)$	add $+3$ to both sides (i.e. add 3)	$x-3+(3)=6+(3)$
Solution		$x+0=2$		$x+0=9$
		$x=2$		$x=9$

• another way to look at the Inverse Operation, using the same equations is:

Properties of Equality

Addition Property
Equation		$x+4=6$		$x-3=4$
Inverse Operation	add $-4$ to both sides (i.e. subtract 4)	$x+4=6$$-4|-4$	add $+3$ to both sides (i.e. add 3)	$x-3=6$$+3|+3$
	simplify	$x+0=2$	simplify	$x+0=9$
Solution		$x=2$		$x=9$

Using Multiplication Property

Properties of Equality

Multiplication Property
Equation		${\frac {x}{6}}=4$		$6x=24$
Inverse Operation	multiply both sides by 6 (isolates x by making the expression $x{\frac {6}{6}}$ which is equal to $x\times 1$)	${\frac {6\times x}{6}}={4}\times 6$	divide both sides by 6 (isolate $x$ by making the expression $x{\frac {6}{6}}$ which is equal to $x\times 1$)	${\frac {6x}{6}}={\frac {24}{6}}$
	cancel $6\div 6$ (because $6\div 6$ $=1$ )	${\frac {{\cancel {6}}\times x}{\cancel {6}}}={4}\times 6$	cancel $6\div 6$ (because $6\div 6$ $=1$ ) (note: ${\frac {24}{6}}$ is the same as $24\div 6$)	${\frac {{\cancel {6}}\times x}{\cancel {6}}}={\frac {24}{6}}$
Solution		$x=24$		$x=4$

Cross-multiplying

• use cross-multiplication to solve for $x$ when $x$ is a denominator (bottom of a fraction)

• see for numerators and denominators

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

Using Cross-Multiplication

${\frac {6}{x}}=8$	is the same as	${\frac {6}{x}}={\frac {8}{1}}$
using cross-multiplication, we can move the variable $x$ to the top of the fraction (numerator)
${\frac {6}{x}}={\frac {8}{1}}$	is the same as (using cross-multiplication)	$8x=6\times 1$
	now we can solve for $x$
$8x=6\times 1$	$8x=16$${\frac {8\times x}{8}}={\frac {16}{8}}$$x={\frac {16}{8}}$	$x=2$

How to solve an equation with two of the same variables

• when an equation has two of the same variables, we isolate the variable by combining its instances
• ex.
  • $2x+5x=35$
  • the values $2x$ and $5x$ may be "distributed" in order to make a single instance of $x$ and thereby allowing for it to be isolated

Distributive property

= the idea that multiplication can be "distributed" through addition

• multiplication is addition by a certain factor (number of times)
• ex. when we multiply $5\times 5$, we are adding 5 five times: $5\times 5=5+5+5+5+5$
  • that is the same as adding $(2+3)$ five times
  • so we can express $(2+3)$ times 5 as either
    • $5\times (2+3)$ or
    • $5\times (5)$ or
    • $(5\times 2)+(5\times 3)$
      • they all equal 25
• with variables, we use the process:
  • $2x+5x$ =
  • $x\times (2+5)$ =
  • $x\times 7$ = 7$x$