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)	$x - 3 = 6$ $+ 3 \| + 3$
Inverse Operation	simplify	$x + 0 = 2$	simplify	$x + 0 = 9$
Solution		$x = 2$		$x = 9$

Using Multiplication Property

Properties of Equality
Multiplication Property
Equation		$\frac{x}{6} = 4$		$6 x = 24$
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 $x \times 1$ )	$\frac{6 x}{6} = \frac{24}{6}$
	cancel $6 \div 6$ (because $6 \div 6$ $= 1$ )	$\frac{6 \times x}{6} = 4 \times 6$	cancel $6 \div 6$ (because $6 \div 6$ $= 1$ ) (note: $\frac{24}{6}$ is the same as $24 \div 6$ )	$\frac{6 \times x}{6} = \frac{24}{6}$
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)	$8 x = 6 \times 1$
	now we can solve for $x$
$8 x = 6 \times 1$	$8 x = 16$ $\frac{8 \times x}{8} = \frac{16}{8}$ $x = \frac{16}{8}$	$x = 2$

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.
- $2 x + 5 x = 35$
- the values $2 x$ and $5 x$ 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 5×5, we are adding 5 five times: 5×5=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
  - (5×2)+(5×3)
    - they all equal 25
with variables, we use the process:
- $2 x + 5 x$ =
- $x \times (2 + 5)$ =
- $x \times 7$ = 7 $x$