Solving Equations

Equations are mathematical statements which occupy a central place in solving problems related to any topic in Math. An equation is a mathematical statement in which two mathematical expressions are related by an equal sign (=). Solving an equation means finding out what number x stands for, if x is a variable.


 Definition of Equation:
An equation is a mathematical statement that tells two algebraic expressions are equal.
 

Solution of an Equation

Solving an equation is the process of finding the roots or solutions of the equation. To solve linear equations, you add, subtract, multiply and divide both sides of the equation by numbers and variables, so that you end up with a single variable on one side and a single number on the other side. When the equation has one variable, the solution is a single number.

Example:
x + 6 = 10, the value of x only make the equation true.

x = 4, because 4 + 6 = 10.

Equation with two variable instead of one. Solution to the equation x + y = 1 will be not a single number but a pair of numbers, one for x and one for y, that makes the equation true.

Properties of an Equality

The following properties of an equality are used in solving an equation

1
. The addition property of equality -

It states that if a = b, then a + c = b + c. The quantity c was added to both sides so that the balance is maintained.

Example: Value of x, x + 1 = 3

Case 1: Solution of x + 1 = 3

x = 3 - 1

= 2   

Case 2: Adding 4 both side, we have,

x + 1 + 4 = 3 + 4

x + 5 = 7

x = 7 - 5

= 2

So, in both the case the value of x is 2.

2. The multiplication property of equality:

It states that if a = b, then a * c = b * c. The quantity c multiplied both sides so that the balance is maintained.

Example: Value of x, x * 2 = 4

then x = 2

if we multiply 3 both side,

then x * 2 * 3 = 4 * 3

x = 2.

  3.  The substitution property of equality:

The substitution property is same as addition property of equality. It states that if a = b, then a - c = b - c. The quantity c was added to both sides so that the balance is maintained.

How to Solve an Equation

An equation is a mathematical statement using numbers, letters and symbols to express a relationship of equality. To solve an equation means to find the numerical values of the unknown variable that makes the equation true.

Steps For Solving the equation:

Step 1: Use the distributive property to separate terms, if necessary.

Step 2: Use addition or subtraction property of equality to get all variable terms on one side of the equation and all constant terms on the other side.

Step 3: Use the multiplicative property of equality to get value of the unknown.

Example:
Solve x + 7 = 15

Using the subtraction property of equality, subtracting 7 from both side of the equation will resulting in altering the given equation.

x + 7 - 7 = 15 -7  
                          
x = 8                    

Transposition Rules

  • If a number is added to a variable or to an expression containing the variable, then subtract the number on the other side.
  • If a number is subtracted to a variable or to an expression containing the variable, then add the number on the other side.
  • If a variable or an expression containing a variable is multiplied by a number, then divide the expression on the other side by the number.
  • If a variable or an expression containing a variable is divided by a number, then multiply the expression on the other side by the number.

Solving One Step Equations

One step equation, an equation that can be solved in one step by using transposition rules.

Solved Examples

Question 1: Find the value of  x,  x + 8 = 18

Solution:
x + 8 = 18

Subtracting 8 from both side, we have

x + 8 - 8 = 18 - 8

x = 10
 

Question 2: Find the value of  y,  y - 3   =  12

Solution:
y - 3   =  12

adding 3 from both side, we have

y - 3 + 3 = 12 + 3

y = 15

 

Question 3:   Find the value of x, 3x = 24

Solution:
3x = 24

Multiplying by $\frac{1}{3}$ from side, we have

3x * $\frac{1}{3}$ = 24 * $\frac{1}{3}$

x = 8.



 

Solving Literal Equations

A literal equation is an equation that contains letters and numbers. To solve a literal equation means to solve for only one of the variables. Solving literal equations is similar to solving other equations. The properties of equality are used to isolate the variable.

Solved Examples

Question 1: Solve this literal equation for  x,  2x + 5y = x + 2y
Solution:
2x + 5y = x + 2y

On subtracting x from both side,

2x + 5y - x = x + 2y - x

(2x - x) + 5y = (x - x) + 2y

x + 5y = 2y

On subtracting 5y from both sides,

x + 5y - 5y = 2y - 5y

x = (2 - 5)y

x = - 3y.

 

Question 2: Solve this literal equation for  p,  x + 2p = 5x - 4p
Solution:
x + 2p = 5x - 4p

On subtracting x from both side

x + 2p - x = 5x - 4p - x

2p = (5x - x) - 4p

2p = 4x - 4p

Adding 4p from both side,

2p + 4p = 4x - 4p + 4p

(2 + 4)p = 4x

6p = 4x

By dividing 6, we have

$\frac{6p}{6}$ = $\frac{4x}{6}$

p = $\frac{2x}{3}$
 

Solving Equations Online

Equations are of many types, like algebraical equations, trigonometric equations.  Online equation solver programs are helpful in solving some of these equations quickly. The equation solver takes the input in a specific form and returns the solutions or solution set.