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(Created page with "== 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...")
 
 
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* ex.:  
* ex.:  
** x + 3 = 8
** <math>x + 3 = 8
* the "inverse operation" adds -3 to both sides of the equation:
</math>
** x + 3 '''- 3''' = 8 '''- 3'''
*the "inverse operation" adds -3 to both sides of the equation:
**<math>x + 3 - 3 = 8 - 3</math>


* which leaves us with
*which leaves us with  
** x + 0 = 5
** <math>x + 0 = 5</math>
** x = 5
**<math>x = 5</math>


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


=== Isolating the variable ===
===Isolating the variable===


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


=== Properties of Equality ===
=== Properties of Equality===


* property = a rule
* property = a rule
* equality = that both sides of the equation (equal sign) have the same value
*equality = that both sides of the equation (equal sign) have the same value
 
=== Variable ===
 
* an unknown value represented, usually represented by the letter <math>x </math>
 
==How to solve an equation with a single variable==
 
===Using Addition Property===
 
*when solving for <math>x</math> when <math>x</math> is added or subtracted to/from another number
**we "isolate <math>x</math>" by using the "Inverse Operation" to remove the number from the side with the variable, <math>x</math>
**note that
***addition is adding a positive number: <math>5 + (+3) = 8</math>
****where <math>(+3)</math> means "positive 3"
***subtraction is adding a negative number: <math>5 + (-3) = 2</math>
****where <math>(-3)</math> means "negative 3"
**
*examples:
 
{| class="wikitable" style="text-align: center;"
|+Properties of Equality
!
!
! colspan="3" |Addition Property
|-
| style="text-align: left;" |Equation
|
|<math>x + 4 = 6</math>
|
|<math>x - 3 = 4</math>
|-
| style="text-align: left;" |Inverse Operation
|add <math>-4 </math> to both sides
 
(i.e. subtract 4)
|<math>x + 4 + (-4) = 6 + (-4) </math>
|add <math>+3 </math> to both sides
 
(i.e. add 3)
|<math>x - 3 + (3) = 6 + (3) </math>
|-
| style="text-align: left;" |Solution
|
|<math>x + 0 = 2 </math>
|
|<math>x + 0 = 9
</math>
|-
|
|
|<math>x = 2 </math>
|
|<math>x = 9
</math>
|}
 
*another way to look at the Inverse Operation, using the same equations is:
 
{| class="wikitable" style="text-align: center;"
|+Properties of Equality
! colspan="5" |Addition Property
|-
| style="text-align: left;" |Equation
|
|<math>x + 4 = 6</math>
|
|<math>x - 3 = 4</math>
|-
| rowspan="2" style="text-align: left;" |Inverse Operation
|add <math>-4 </math> to both sides
 
(i.e. subtract 4)
|<math>x + 4  =    6
  </math><math>- 4  |      -4 </math>
|add <math>+3 </math> to both sides
 
(i.e. add 3)
| style="bottom-border: none;" |<math>x - 3 =    6 </math><math>+ 3  |    +3 </math>
|-
|simplify
|<math>x + 0 = 2 </math>
|simplify
|<math>x + 0 = 9
</math>
|-
|Solution
|
|<math>x = 2 </math>
|
|<math>x = 9
</math>
|}
 
=== Using Multiplication Property ===
{| class="wikitable" style="text-align: center;"
|+Properties of Equality
! colspan="5" |Multiplication Property
|-
|Equation
|
|<math>\frac x 6 = 4 </math>
|
|<math>6x = 24 </math>
|-
|Inverse Operation
|multiply both sides by 6
(isolates x by making the expression <math>x \frac 6 6 </math>  which is equal to <math>x \times 1 </math>)
|<math>\frac {6\times x} 6 = {4} \times 6  </math>
|divide both sides by 6
(isolate <math>x </math> by making the expression <math>x \frac 6 6 </math>
which is equal to <math>x \times 1 </math>)
|<math>\frac {6x} 6 = \frac {24} 6  </math>
|-
|
|cancel <math>6 \div 6
</math>
(because <math>6 \div 6
 
</math> <math>= 1
 
</math> )
|<math>\frac {\cancel6\times x} \cancel 6 = {4} \times 6  </math>
|cancel <math>6 \div 6
</math>
(because <math>6 \div 6
 
</math> <math>= 1
 
</math> )
 
(note: <math>\frac {24} 6  </math> is the same as <math>24\div 6  </math>)
|<math>\frac {\cancel6\times x} \cancel 6 = \frac {24} 6  </math>
|-
|Solution
|
|<math>x = 24
</math>
|
|<math>x= 4
</math>
|}
 
=== Cross-multiplying ===
 
* use cross-multiplication to solve for <math>x  </math> when <math>x  </math> is a denominator (bottom of a fraction)
 
* see for numerators and denominators
 
==== How to solve for <math>x  </math> when <math>x  </math> is a denominator: "Cross-multiplication" ====
{| style="text-align: center;"
|+Using Cross-Multiplication
!<math>\frac 6 x = 8  </math>
!is the same as
!<math>\frac 6 x = \frac 8  1 </math>
|-
| colspan="3" |using cross-multiplication, we can move the variable <math>x  </math> to the top of the fraction (numerator)
|-
!<math>\frac 6 x = \frac 8  1 </math>
!is the same as
(using cross-multiplication)
|<math>8x= 6 \times 1  </math>
|-
|
|now we can solve for <math>x</math>
|
|-
|<math>8x= 6 \times 1  </math>
|<math>8x= 16  </math><math>\frac {8\times x} 8 = \frac {16} 8  </math><math>x = \frac {16} 8
  </math>
|<math>x=2
 
  </math>
|}
 
== 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.
** <math>2x + 5x = 35</math>
** the values <math>2x
</math> and <math>5x
</math> may be "distributed" in order to make a single instance of <math>x
</math> 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 <math>5 \times 5</math>, we are adding 5 five times: <math>5\times 5 = 5+5+5+5+5
</math>
** that is the same as adding <math>(2+3)</math> five times
** so we can express <math>(2+3)</math> times 5 as either
*** <math>5 \times (2+3)
 
</math> or
*** <math>5 \times (5)
 
</math> or
*** <math>(5\times 2) + (5 \times 3)
</math>
**** they all equal 25
* with variables, we use the process:
** <math>2x + 5x</math> = 
** <math>x \times (2+5)
</math> =
** <math>x \times 7
 
 
</math> = 7<math>x
</math>


[[Category:Math]]
[[Category:Math]]
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