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How to Solve Linear Equations (Application and Techniques)


How to Solve Linear Equations: Would learning how to solve linear equations interest you? The act of solving a linear equation is known as finding its solution. We covered the methods for solving linear equations in this article.

How to Solve Linear Equations (Application and Techniques)

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).

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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:

▸ AX = B ————- (i)

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

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