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Solving Linear Equations with Two Variables

A system of linear equations in two variables is a set of equations of degree one defined in two variables.
For example: 3x + 5y = -1 and -x + 2y = 4.

Solving linear equations in two variables means to find the values of both variables. There are various methods of solving them. These methods are as follows:

  • Substitution Method:  According to this method, find the value of one variable from one equation in terms of other variable and substitute it in other equation. Find values of both the variables.
  • Elimination Method: In elimination method, we equate the multiples of a particular variable in both equation and then add or subtract the equations to eliminate that variable which results in finding the value of other variable. Determine the value of remaining variable in the same way.
  • Graphical Method: According to graphical method, we need to draw the graph for both the equations which will be straight lines. Note down the point of intersection of both lines. This would be the required solution.
  • Matrix Method: Matrix method is easy but lengthy method to solve the linear equations in two variables. We use the following identity:
Solving System of Equations
Where, A and B are matrices of Multiples of variables and constant terms respectively.

Let us understand an example of solving linear equations in two variables.

Example: Solve the following system of linear equations in two variables by substitution as well as elimination method-
2x + 3y = 5
x + y = 1

Solution: Let us assign numbers to given equations
2x + 3y = 5 .......(1)
x + y = 1...........(2)

Substitution Method
Find the value of y from equation (2),
y = 1 - x .........(3)
Substitute it in equation (1),
2x + 3 (1 - x) = 5
2x + 3 - 3x = 5
x = - 2
Substituting the value of x in equation (3),
y = 1 - (- 2)
y = 3
Hence, the solution is x = - 2 and y = 3.

Elimination Method:
Multiplying equation (2) by 2, we get
2x + 2y = 2 .........(4)
Subtracting equation (4) from equation (1), we obtain:
y = 3
Multiplying equation (2) by 3, we get
3x + 3y = 3 .........(5)
Subtracting equation (5) from equation (1), we obtain:
-x = 2
x = -2
Hence, the solution is x = -2 and y = 3.

Related Calculators
Linear Equations in Two Variables Solver Solving Two Step Equations
Solving Equations with Variables on both Sides Calculator Graphing Inequalities in Two Variables Calculator
 

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