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Algebra: Difference between revisions
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→Properties of Equality
(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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| Line 23: | Line 23: | ||
* ex.: | * ex.: | ||
** x + 3 = 8 | ** <math>x + 3 = 8 | ||
* the "inverse operation" adds -3 to both sides of the equation: | </math> | ||
** x + 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 === | ===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 | ||
==How to solve an equation== | |||
===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> | |||
|} | |||
{| class="wikitable" | |||
|+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> ) | |||
|<math>\frac {\cancel6\times x} \cancel 6 = {4} \times 6 </math> | |||
|- | |||
|Solution | |||
| | |||
|<math>x = 24 | |||
</math> | |||
| | |||
|<math>x= 4 | |||
</math> | |||
|} | |||
asdf | |||
[[Category:Math]] | [[Category:Math]] | ||