how to solve linear equations

How to Solve Linear Equation In Efficient Manner

Linear equation is a mathematical term and is an equation which is put into form and contains the variable, coefficient which often real numbers. The parameter of the equation contains coefficients which may have arbitrary expressions and it does not reflect any variables. There is also another way though which linear equation can be obtained. Another way is equating zero as a linear polynomial over some field through which a linear equation. Where the solution of such an equation can be proved true by putting some value. The substitute value becomes equal and the solution of such an equation becomes true.

Linear equation contains several variables such as there may be one variable, two variables and more than two variables.

Solving the linear equation containing One Variable:

Solving the linear equation indicates that both sides of the equation should be equal.  One side of the linear equation should be equal to another side of the linear equation.  We need to use certain mathematical operations on both sides so that such can be equal.


Question: 2x – 12/2 = 3(x – 1)

First Step: Fraction should be clear

2x – 6 = 3(x – 1)

Step 2: Simplification of equation on both side

2x – 6 = 3x – 3

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=>2x = 3x + 3

=>2x – 3x = 3

Step 3: Isolate x

x = -3

Linear equations can be understood implicitly as one variable. You can understand through given equation:

ax b =0, the unique solution come from the equation is x = -b/a

Generally, in above solution a is not equal to zero, the variable x is sensible unknown. However, where the a = 0, there may be two cases, either every number is solution or b is also equal to 0. If b is not equal to 0, there is no solution and in this case the equation becomes inconsistent.

Solving the linear equation containing Two Variable:

The linear equation containing two variables can be present in the given format:

ax+ by+ c=0, in the given case, there are two variables which are x and y and coefficients are a, b and c.

The linear equation containing two variables can be solved by using different methods. Some of them are as given below:

(a) Substitution method: in this method, two variables are substituted by one variable to solve the equation.

(b) Cross multiplication variable: In this method, two variables are crossed to simplify the equation to determine the value of variable a.

(c) Elimination method:  In this method, linear equations can be solved by eliminating one variable out of two by using mathematical operations with the same value.

(d) Determinant Method: In this method, the linear equation can be solved by using denominator and numerator determinant to replace the variable.

Solving of linear equations containing three variables:

Solving the linear equation under three variables, three sets of equations is required to find the unknown values.  The linear equation containing three equations can be solved by using the matrix method which seems very popular. Under matrix method, linear equation can be solved by using the following steps:

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(a) There should be appropriate order in respect of all variables in the given equation.

(b) There should be proper attention that the variable, coefficient and constants are shown on their respective side.

Where the equation consists of the method of finding the inverse consists of two new metrics namely Matrix R- it indicates variable and Matrix P: it indicates constant. Such systems of equations can be solved by using matrix multiplication.

Method for Solving the linear equation is as follows:

Using the least common denominator to clear the fraction if the equations contain any fraction. This can be done by multiplying both sides of the equation by LCD.

Also, if the variables are present in the denominator of the fractions, the value of such variables should be identified which will give divisions by zero as this value needs to be avoided in our solutions.

Equation should be simplified by both sides.

First two facts given above should be used to get all terms with the variable in them on one side of the equation and on the other side all constant.

Use the third or fourth fact to make the coefficient a one if the coefficient of the variable is not a one.

The final step as it is the most skipped step that is verifying the answer. Whether or not you have got a correct answer this step helps to determine it. Answer is verified by plugging the result from the previous steps into the original equation. Since you may have made mistakes in very first steps which leads to incorrect answers it is very important to plug into the original equation.

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Note: If in the problem there were fractions and also values of variables that give division by zero it is important to make sure that one of these values does not end up in the solution set.

General steps for solving linear equations:

  1. It needs to expand the problems showing all the brackets.
  2. There should be rearrangement of variables and constants on each side of the linear equation.
  3. The equation should be simplified by making grouping of variables and constant.
  4. Factorization needs to be done as required.
  5. Once, all the above steps follow, Solution will be complete.
  6. Answers will be checked by substituting the solution back into the original equation.


The linear equation is complex to solve but it can be done easily through following the above steps and applying the necessary mathematical operation. As discussed above, different methods should be applied in respect of each different linear solution. Get the best math assignment help from the experts at nominal charges.

Frequently Asked Questions

What is the golden rule for solving equations?

When solving numerical equations, we should balance both sides of the equations so that both sides are always equal.

What is a true linear equation?

It looks like any other equation. It consists of two expressions set equivalent to one another. It can also be written in the form of ax+b-=0. Linear equations graph as straight lines.

What are the three types of equations?

Three major forms of linear equations: point-slope form y-y*=m(x-x*), standard form Ax+By=c, and slope-intercept form y=mx+b.

How do you write an equation for a linear function?

In the equation f(x)=mx+b if we use m=0 , the equation simplifies to f(x)=b. In other words, the value of the function is a constant.

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