Algebra/Solving Equations

Equations

Introduction

An equation is a mathematical statement of the equality of two quantities. For example, the equation $x=4$ is a statement which says that the quantity $x$ is equal to 4. Similarly, the statement "the square of the sum of a number and 6 is equal to 49" can be expressed as the equation $(x+6{)}^2=49$. One more example: "the sum of the square of a number and 6 is equal to 7" can be written as $x^2+6=7$.

Solving for (Variable)

When solving an equation, you usually solve for a specific variable. To do so, you have to get all instances of that variable on one side of the equals sign, and everything else on the other.

Properties of Equality

The equal sign that depicts the fact that both sides of it are equal is a very strange symbol with many properties. It tells you various traits of each side, and it allows you to manipulate each side in specific ways. Here are the different properties of that sign:

Property Name	Definition	Example
Reflexive	a = a	7 = 7
Symmetric	If a = b, then b = a	If (3)(2) = 6, then 6 = (3)(2)
Transitive	If a = b & b = c, then a = c	If 8 = (4)(2) and (4)(2) = (2)(4), then 8 = (2)(4)
Substitution	If a = b, then one can replace a with b or vice versa	If a = b and 1 + a = 3, then 1 + b = 3
Addition	You can add one number to both sides of the equation.	$x-6=14$ $x-6+6=14+6$ $x=20$
Subtraction	You can subtract one number from both sides of the equation.	$x+6=14$ $x+6-6=14-6$ $x=8$
Multiplication	You can multiply both sides of the equation by a number.	$\frac{x}{6}=18$ $6(\frac{x}{6})=(18)(6)$ $x=108$
Division	You can divide both sides of the equation by a number.	$6x=18$ $6x(\frac{1}{6})=(\frac{18}{6})$ $x=3$

Practice Problems

Decide whether these following problems are expressions or equations.

1. $2x+6$

2. $3(x-14{)}^2=12$

3. $4+6(2x+16)$

Identify which properties are being used in the following problems.

1. $a=b$ and $3-a=4$, so $3-b=4$.

2. $x+9=12$, then $x=3$.

3. $x-9y=4y$, then $x=13y$.

Answers

1. Expression

2. Equation

3. Expression

1. Substitution Property of Equality

2. Subtraction Property of Equality

3. Addition Property of Equality (combine like-terms!)

Basic Laws In Algebra

There are several basic laws in algebra. Understanding these will help you to manipulate and solve equations, and to understand algebraic relationships.

1. Commutative Law -

General - The order of the items can be changed without affecting the results.

For addition A + B = B + A indicating that changing the order of the items added does not affect the sum.

For multiplication X Y = Y X indicating that the changing of the order of the items multiplied does not affect the product.

2. Associative Law –

General - The grouping of the items can be changed without affecting the results. (Seems to be an extension of the commutative law).

For addition A + (B + C) = (A + B) + C indicating that changing the grouping of the items added does not affect the sum.

For multiplication X ( Y Z ) = ( X Y ) Z indicating that the changing grouping of the items multiplied does not affect the product.

3. Distributive Law -

Indicates that common terms can be factored, or that factors can be distributed. (A + B) X = (A X) + (B X) (The "X" terms on the right are combined into a factor on the left side; the factor "X" on the left is "distributed on the right side).

Consider the substitution of X = (Y + Z) into the above equation yields (A + B) (Y + Z) = A (Y + Z) + B (Y + Z). Apply the distributive law to each term on the right yeilds A Y + A Z + B Y + B Z We can skip the intermediate step if we multiply the terms identified by “F O I L” in the following expression (A + B) (Y + Z) =

F - first terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A Y +

O - outside terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A Z +

I - inside terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B Y +

L - last terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B Z

4. Law of Identity -

For addition and subtraction the law of identity indicates that the addition and subtraction of a given term or quantity results in the zero, 0, the identity element for addition and subtraction. Alternately, adding the identity element results in in no change to the original value or quantity.

$A-A=0$

Adding A to both sides of the first equation we get (A - A) + A = 0 + A. Re-arranging or substituting gives 0 + A = A

 Note the special case(s) where A = A + 0 = A + 0 + 0

For multiplication and division the law of identity indicates that the multiplication and division of a given term or quantity results in "one", 1, the identity element for multiplication and division. Alternately, multipling or dividing by the identity element results in in no change to the original value or quantity.

$1=\frac{Y}{Y}$, or $1=(\frac{Y}{1})(\frac{1}{Y})$

 Note that dividing 1 by a term or quantity gives the 
 reciprocal of the term or quantity.  Multiplying by the 
 reciprocal is the same as dividing by the term or quantity.
 In the above equation on the right (Y / 1), and (1 / Y) are
 reciprocals of each other

 Note the special case where $1=\frac{1}{1}$,
 Multiplying this equation by “1” gives  $1(1)=(1)(\frac{1}{1})$ ,
 and then dividing by one gives
 $\frac{1(1)}{1}=(1)(\frac{1}{1})=$.
 
 Simplify this by substititing the first special case
 equation to get $1=1(1)$ , and $1=1(1)(1)$,  . . .

By multiplying both sides of the first equation by “Y” we get $(Y)(1)=(Y)(\frac{Y}{Y})$ , which simplifies and becomes (Y) = (1) Y.

5. Proportions or Ratios -

Ratios or proportions are sometimes written as an equation of fractions

(for example $\frac{W}{4}=\frac{3}{6}$ ),

or they can be expressed as a relationship Q : R = S : T , (expressed in words “ ‘Q’ is to ‘R’ as ‘S’ is to ‘T’ ”).

One needs to be careful since each term also has a proportional relation with the two another variables.

 In our examples all of the following are also valid
 Q : S = R : T ,“ ‘Q’ is to ‘S’ as ‘R’ is to ‘T’ ”.
 R : Q = T : S ,“ ‘R’ is to ‘Q’ as ‘T’ is to ‘S’ ”..  
 S : Q = T : R ,“ ‘S’ is to ‘Q’ as ‘T’ is to ‘R’ ”.
   
 W : 4 = 3 : 6,  and  W : 3 = 4 :6 ,
 4 : W = 6 : 3,  and  3 : W = 6 : 4.

Consider the relation described by the equation $(A)(B)=(Y)(Z)$

Dividing each side of the equation by $(B)(Z)$ gives

$\frac{(A)(B)}{(B)(Z)}=\frac{(Y)(Z)}{(B)(Z)}$ , which simplifies to $\frac{(A)}{(Z)}=\frac{(Y)}{(B)}$

If we divide by $(B)(Y)$ instead the results simplify to

$\frac{(A)}{(Y)}=\frac{(Z)}{(B)}$

Dividing each side of the first equation by $(A)(Z)$ gives

$\frac{(A)(B)}{(A)(Z)}=\frac{(Y)(Z)}{(A)(Z)}$ , which simplifies to $\frac{(B)}{(Z)}=\frac{(Y)}{(A)}$

If we divide by (A)(Y) instead the results simplify to

$\frac{(B)}{(Y)}=\frac{(Z)}{(A)}$

Solving Equations

Although we have already solved a few equations, we will now discuss the formal idea of solving equations. To solve an equation, you are finding the value of any variables within the equation. To find the value of a variable, you have to manipulate the equation to state $*insertvariablehere*=*somenumber*$. Then you know the value of the variable! You will use the Properties of Equality to manipulate the equation into the desired form.

Practice Problems

Solve for x in the following equations.

1. $x+7=12$

2. $-\frac{2}{3}x+9=15$

3. $3x-15=8x$

4. $c(x-4m)=d(n-9)$

5. $20x+12=4(5x+3)$

Answers

1. $\begin{array}{ccc}x+7& =& 12\\ x& =& 5\end{array}$ Subtract 7

2. $\begin{array}{ccc}-\frac{2}{3}x+9& =& 15\\ -\frac{2}{3}x& =& 6\\ x& =& -9\end{array}$ Subtract 9, then multiply by -3/2 (Inverse Property of Multiplication)

3. $\begin{array}{ccc}3x-15& =& 8x\\ -15& =& 5x\\ -3& =& x\end{array}$ Subtract 3x, then divide by 5

4. No Solution, because there are too many variables to find a single number for x.

5. $\begin{array}{ccc}20x+12& =& 4(5x+3)\\ 20x+12& =& 20x+12\end{array}$ All Real Numbers. Perform Distributive Property, and you'll get the same equation on both sides. Thus, any number would work!

Lesson Review

Equations are two expressions that are equal to each other, and they are expressed by putting each of them on one side of the equal sign. You can add, subtract, multiply, or divide both sides of an equation while keeping it equal (For example, we know that 7 = 7, correct? What if we subtracted 2 from each side? We'd still have a true statement: 5 = 5). There are other properties of equality, such as the Reflexive, Symmetric, Transitive, and Substitution. You will be using all of these properties to solve (find the value of) variables in equations.

Lesson Quiz

1. What property is expressed here? $a=b$ and $b=c$, then $a=c$.

2. If I divided both sides of an equation by 4, would it still be equal on both sides? If so, why?

3. Solve for y. $5(\frac{y}{6}-2)=2y+4$

Quiz Answers

1. Transitive Property of Equality

2. Yes, due to the Division Property of Equality.

3. y=-12