How to Solve Linear Equations (Application and Techniques)

Would you be interested in finding out how to solve linear equations? Finding a linear equation’s solution is the definition of solving a linear equation. In this article, we discussed the procedures for solving linear equations.

how to solve linear equations

How to Solve Linear Equations

Finding the solution to linear equations in one, two, three, or more variables is referred to as solving a linear equation.

The value or values of the variables included in the equation are referred to as the solution of a linear equation.

To solve linear equations, there are six primary approaches. These techniques for solving linear equations include:

1. Graphical Method

2. Elimination Method

3. Substitution Method

4. Cross Multiplication Method

5. Matrix Method

6. Determinants Method

Graphical Method of Solving Linear Equations

You must first graph both equations in the same coordinate system in order to solve linear equations graphically.

Next, look for the intersection point on the graph. Take the equations 2x + 3y = 9 and x – y = 3, for instance.

Consider x = 0; 1, 2, 3, 4; and then solve for y to draw the graph. Plot the points on the graph as soon as (x, y) has been determined.

We should notice that the graph will be more accurate if there are more x and y data.


Elimination Method of Solving Linear Equations

Any coefficient is first equal and then removed in the elimination technique.

The equations are solved to provide the other equation after elimination. For better comprehension, they show the elimination approach for solving linear equations below.

Check these equations

2x + 3y = 9 — (i)


x – y = 3—- (ii)

Here, the coefficient of “x” will become the same and may be removed if equation (ii) is multiplied by two.

Therefore, multiply equation (ii) by 2 before removing equation (i)

2x + 3y = 9


2x – 2y = 6

-5y = -3

Or, y = ⅗ = 0.6

Add the value of y = 0.6 to equation now (ii).

So, x – 0.6 = 3

Thus, x = 3.6

Thus, the values of x and y are discovered to be 3.6 and 0.6, respectively.

Substitution Method of Solving Linear Equations

To use the substitution method to solve a linear equation, you must first determine the value of one variable from each equation.

The second equation can then be solved by substituting the value of the isolated variable. Consider the same equations once more.


2x + 3y = 9 I (i)


x – y = 3 ———- (ii)

Consider equation (ii) now, focusing on the variable “x.”

Equation (ii) then becomes

x = 3 + y.

Change the value of x in the equation now (i). The equation will thus be:

2x + 3y = 9

▸2(3 + y) + 3y = 9

▸6 + 2y + 3y = 9

Or, y = ⅗ = 0.6

Now, replace the value of “y” in equation (ii).

x – y =3

▸ x = 3 + 0.6

Or, x = 3.6

thus (x, y) = (3.6, 0.6).

Cross Multiplication Method of Solving Linear Equations

Cross multiplication is an efficient way to solve linear equations. The cross-multiplication method is applied in this approach to streamline the answer.

The following formula is used to solve two-variable equations via cross-multiplication:

x /(b1 c2 − b2 c1) = y / (c1 a2 − c2 a1) = 1 /(b2 a1 − b1 a2)

For example, consider the equations

2x + 3y = 9 —(i)


x – y = 3 —(ii)


a1 = 2, b1 = 3, c1 = -9

a2 = 1, b2 = -1, c2 = -3

Now, solve using the aforementioned formula.

x = (b1 c2 − b2 c1) / (b2 a1 − b1 a2)

Putting the respective value we get,

x = 18/5 = 3.6

Similarly, solve for y.

y = (c1 a2 − c2 a1) / (b2 a1 − b1 a2)

So, y = ⅗ = 0.6

Matrix Method of Solving Linear Equations

The matrix technique may also solve linear equations. For the solution of linear equations involving two or three variables, this approach is quite useful. Think of these three equations as:

a1x + a2y + a3z = d1

b1x + b2y + b3z = d2

c1x + c2y + c3z = d3

These equations have the following form:

linear equations

▸ AX = B ————- (i)

The X matrix, A matrix, and B matrix are:

how to solve equations

Now, multiply (i) by A-1 to get:

A−1AX = A−1B ⇒ I.X = A−1B

▸ X = A−1B

Determinant Method of Solving Linear Equations (Cramer’s Rule)

It is simple to solve linear equations in two or three variables using the determinants technique.

The process is as follows for linear equations with two and three variables.

For Linear Equations in Two Variables:

x = Δ1/Δ,

y = Δ2/Δ

Or, x = (b1 c2 − b2 c1) / (b2 a1 − b1 a2) and y = (c1 a2 − c2 a1) / (b2 a1 − b1 a2)


Methods of Solving Linear Equations in One Variable

A linear equation with only one variable may be solved quickly and easily.

Bring all the terms for the variable on one side and the constants on the other to solve any two equations with just one variable.

The equation’s solution can also be found using the visual approach, where the point where the line intersects the x- or y-axis is the indicator.

Take the equation 2x + 4 + 7 = 4x – 3 + x, for instance.

Here, combine the “x” words and bring them to one side.


5x – 2x = 14

Or, x = 14/3

Methods of Solving Linear Equations in Two Variables

Any of the methods listed above, the cross multiplication method, matrix method, and determinants method, can solve a linear equation in two variables.

Methods of Solving Linear Equations in Three or More Variables

The graphical, elimination and substitution approaches cannot be used to solve every problem with three or more variables.

The cross-multiplication method is the most popular approach for resolving equations with three variables.

When trying to solve problems with three or more variables, even the matrix Cramer’s rule comes in quite handy.

To solve an equation, you must identify the value of the variable that makes it true.

When solving an equation, you often want to reverse all operations done to the variable in order to get it back by itself.

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CSN Team.

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