Linear Equations with Two Variables

 An equation comes known as a linear equation with two variables only when the variables are noted down as ax + by + c = 0. Here, a, b and c are referred to as real numbers and they are also termed as coefficients of x and y so a and b should not be equal to 0. For example: 4x +10y = 3 and -5x + y = 2 these are linear equations with two variables. 

A linear equation is a system that can be solved by 3 types of solution–unique solution, no solution, and infinitely many solutions. The equation of the linear equation of two variables is represented by y = MX + b. Here, m refers to the slope and b refers to the intercept of y. 

Methods of solving linear equations in two variables:

5 methods help in solving linear equations in 2 variables:

• Graphical method 

The steps that lead to solving of linear equation with 2 variables are:

1. To evaluate the two variables of the two equations graphically. 
2. To manually graph the equation and then transform it into y = MX +b form to find the value of y first. 
3. Then we should put the value of x which is given, which helps in finding the value of y and vice versa.
4. They find out the point where both the lines intersect. 
5. The point at which both the lines meet is the solution. 

• Substitution method

The steps are:

1. Firstly, evaluate one of the 2 equations. 
2. Then substitute that in the 2nd equation for having an equation with a single variable.
3. Evaluate the variable. 
4. To put in any of the equations which will help in finding out another variable’s value.

• Cross multiplication method

The cross-multiplication method means multiplying in a cross direction, so the left upward equation will be multiplied by the right downward equation and the left downward equation will be multiplied by the right upward equation. Then, the equation should be solved and the variables should be calculated.

• Elimination method

The steps are:

1. Bring the equation into a proper form. 
2. To make sure that the addition or subtraction of the equation should not lead to the cancellation of any variable.
3. Then we should use the coefficient of either x or y and put it in the equation and make sure that this leads to the cancellation of the variables. 
4. Solve the equation of a single variable.
5. Then send the value to another equation to find out the value of another variable.

• Determinant Method

The steps are:

1. First, to know the determinant which is created by the coefficient of x and y and name it as ∆
2. Here we can find the ∆x which is gained by substituting the 1st column ∆ with constant and it is the same for ∆y 

Practical Applications:

Application of linear equation is: 

Find out the value of the variables of the equation:

2x + 5y = 20 and 3x +6y = 12.

Answer:

By using the substitution method, 

We should multiply the 1st equation with 3 and the 2nd equation with 2. So,

6x + 15y = 60

6x +12y = 24 

3y = 36 which is y = 12. 

Let’s put the value of y in 1st equation so:

2x + 5*20 = 20 

x = -20. 

Conclusion:

As we can conclude that solving linear equations with 2 variables can be done by 5 methods as explained, so it becomes very easy to solve as we are having different methods for the same problem. Then as there is an ‘n’ number of variables so it does not make it difficult for us to solve as along with the ‘n’ number of variables, there is also an ‘n’ number of the equation to solve. Practical application of a linear equation is very important as it helps us to get proper knowledge on how to solve the sum and this is widely used.