Solving Equations with Integers

Equations are a significant part of every mathematical concept. Solving equations with integers is a way to solve mathematical equations using numbers. Compared to other equations, it is exceptionally easy to solve equations with integers as the variable is isolated on one side with other values on the other side. Addition and subtraction properties are used to move the variable term, whereas multiplication or division properties are used to determine the variable value. 

This article delves into solving equations with integers. You will find a brief description of the concept of equations in maths, a thorough explanation of solving equations with variables on both sides, examples, and so on. 

What is an equation in mathematics? 

In mathematics, an equation is an expression that equals two different values. When the = sign is present, the value on the left-hand side should be the same as the value on the right-hand side of the equation. In a linear equation, there is either one variable or more than one variable. 

Here is an example of the equation: x = 25 +15; find the value of x. 

Looking at the equation, it can be said that the value of x needs to be equal to the number 40, as 25 + 15 = 40. 

Here is the standard form used for representing an equation: 

Ax + By = C 

According to this equation, we need to find both x and y.

What are integers? 

Integers can be described as a collection of whole and negative numbers. Same as whole numbers, integers do not have a fractional component. Hence, integers can be positive, negative, or even zero; however, they cannot be in the form of a fraction. All arithmetic operations can be performed on integers, including subtraction, addition, multiplication, and division. Some examples of integers are 1, 2, 5, 8, -9, and -12. Integers are denoted as “Z”. 

There are mainly three types of integers: 

• Positive integers 
• Negative integers 
• Zero 

Solving equations with integers 

As compared to other equations, solving equations with integers is exceptionally easy. They can be solved by isolating the variable on one side with other values on the other side. Here are the steps for solving equations with integers: 

1. The first step is to simplify each side if required.
2. The next step is to use addition or subtraction properties to move the variable term to one side and all other terms to the other side.
3. The next step is to use multiplication or division to determine the variable value.
4. The last step is to check your solution.

Now, let’s solve an example following the above-mentioned steps.

Example: Solve 6x – 32 = 8 – 2x

Solution:

Equation: 6x – 32 = 8 – 2x

Step 1: transfer variables to one side of the equation

6x + 2x = 8 + 32

Step 2: solve by adding or subtracting:

6x + 2x = 8 + 32

8x = 40

Step 3: divide the equation by 8 on both sides

8x = 40

8x/8 = 40/8

x = 5

Step 4: verify the answer by putting the value of x in the equation:

6x – 32 = 8 – 2x

6*5 – 32 = 8 – 2*5

30 – 32 = 8 – 10

-2 = -2

The left-hand side is equal to the right-hand side; hence the value of x that we get solving the equation is correct.

Explaining linear equations 

A linear equation can be described as an equation with the highest degree of power of one variable. In simple terms, no variable has an exponent of more than one in a linear equation. In a graph, a linear equation constantly forms a straight line.

An example of a linear equation in which only one variable is present is Mx + N = 0, where M and N are constant, and x is a variable. An example of a linear equation in which two variables are present is Mx + Ny = O, where M, N, and O are constant and x and y are variables.

There are three ways to write the standard form of linear equations.

1. In one variable: px + q = 0, where p and q ≠ 0 are integers; x is a variable.
2. In two variables: px + qy = r, where p, q, and r ≠ 0 are integers; x and y are variables.
3. In three variables: px + qy + rz = s, where p, q, r, and s ≠ 0 are integers; x, y, and z are variables.

Conclusion 

With this, we come to an end to solving equations with integers. To recap, integers are a collection of whole and negative numbers. Same as whole numbers, integers do not have a fractional component. Hence, integers can be positive, negative, or even zero. When solving equations with integers, we learnt that addition and subtraction are used to move the variable term, whereas multiplication or division are used to determine the variable value. The following is a detailed explanation of the steps for solving equations with integers.