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Solving Equations with integers

Integers are defined as a set of {- infinity, -2, -1, 0, 1, 2, … + infinity}. Hence, no rational or irrational numbers are part of integers and equations involving integers are those equations which require properties of integers to solve them.

Method to Solve Equations with Integers: To solve an integer equation, we can adopt any one of the following method:

1)    Like Terms Aside: In this method, we have to make that unknown variable independent by taking that variable on one side of the equation (generally on Left hand side) and the other integers on the other side of the equation (Right Hand Side).
In the end, we will solve for the unknown variable on Left hand side to get answer on Right Hand Side.

2)    Additive Inverse: Other way of solving equations with integers, is to add the opposites of the integers given to both the sides.
As we know, that the sum of a number and its opposite is always zero, therefore, we can applying this rule, where the opposite of a number is also known as the additive inverse of the number.

Types of Equations:  Equations may contain integers with different pattern, which requires one or more steps before arriving at its solutions.

An equation that needs only one step or operation to solve is known as one-step equation.
To solve one–step equation, we just do the opposite of whatever operation is involved.

Note that for solving equations, it must be remembered that whatever is done on one side of the equation must be done on the other side, as well. Also, we apply the rules for performing operations with integers.

When solving two step equations, which are generally of the linear form: px + q = r or px − q = r, we will solve px + q = r, by subtracting q from both sides of the equation and then divide both sides by p.

While we are solving equations form of "px − q = r, we will have to add q to both the sides of equation and thereafter we need to divide both the sides by p.
In short, we can now start by dividing both the sides by p instead and thereafter subtract and or add something to both sides. We still get the same answer.

Solved Examples:
Example 1: Solve for x, where x – 8 = 15
Solution: Now, the variable x is subtracted by a constant 8, we will do the opposite which is to add 8 to both sides of the equation. Therefore, the unknown variable of x is 23.

Example 2: Solve for 3x – 8 + 5 = - 5x + 6 – 8x.
Solution: By applying the first method, we will combine like terms on left hand side as, 3x + 5x + 8x = 8 - 5 + 6, this implies 16x = 9, that is
x = $\frac{9}{16}$ is the answer.

Example 3: Solve for x + 9 = 18.
Solution: In this case, it is a one step equation and can be solved by applying additive inverse property that is by subtracting 9 from both sides, we will get, x = 9 as the answer.

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