How To Solve Complex Equations In One Variable

Solutions to Complex equations in one variable

When equations in one variable are containing multiple operations, such as parenthesis or division, they become complex equations. To solve complex equations in one variable, you should know the order of operations and combing like terms very well.

If you are reading this article as my first article then you can read my articles on order of operations and combining like terms to understand both the math concepts.

As you already know that the solving equations in one variable is basically isolating the variable at one side of the equation (I prefer to have variable by itself on the left hand side of the equation). Once you isolate the variable on left hand side (or you can choose to isolate it on the right hand side), you solved your equation.

Following is an example of complex equation in one variable "n" and detailed explanations on the solution for the equation.

Problem: Solve for "n"

2(3n - 4) - (7 - 5n) + 8 = 9 - 8(n - 1) + 17n

Solution:First of all start your solution by rewriting the problem in first step as shown below:

2(3n - 4) - (7 - 5n) + 8 = 9 - 8(n - 1) + 17n

Now analyze all the operations involved in the equation. In this equation, there are brackets (parenthesis) involved, and as you know brackets have the first priority to solve over all the other operations, hence open the brackets on both sides in your second step as shown below:

6n - 8 - 7 + 5n + 8 = 9 - 8n + 8 + 17n

I underlined term just to explain for the solution of the brackets after opening them and you need not to underline terms in your solutions.

In step three combine the like terms at both sides of the equation.

n(6 + 5) - 8 - 7 + 8 = n(- 8 + 17) + 9 + 8

I just rewrite all the terms on both sides of the equation by pulling common "n" to combine like terms and by writing all the numbers (constant terms) together.

Now, we solve all the terms as; n(6 + 5) = 11n and - 8 - 7 + 8 = - 7 on left hand side. On right hand side; n(- 8 + 17) = 9n and + 9 +8 = 17. So, our equation becomes as below:

11n - 7 = 9n + 17

Notice that, we have variable terms (11n and 9n) on the both sides of the equations and so do the constant terms (- 7 and 17). So, rearrange the terms to have both variable terms on the left hand side and both constant terms at right hand side of the equation.

Remember, this method of moving the terms to the other side of equation is an imaginary method and works very well for solving equations.

Move 9n to left hand side (LSH) and change its positive sign (not shown at its front) to negative and move - 7 to RHS and change its minus sign to plus as shown in the next step.

11n - 9n = 17 + 7

2n = 24

Now, take 2 (which is multiplying with variable "n") to RHS and divide 24 by 2.

(Actually we divided by "2" on both sides, but in complex algebraic equations "moving the terms from one side to other method" works better though it is an imaginary method)

n = 24/2

We got a fraction as our answer; reduce it into its lowest terms if possible. As 24/2 is 12, hence the answer is "n = 12".

Follow the same procedure to solve your complex equations in one variables and I hope you might be successful in solving them easily.