Methods to Solve System of Equations

A system of linear equations refers to a set of linear equations, which involves one or more same variables. To solve a system of linear equations involving n variables, there needs to be at least n number of equations present in that particular system. There are several methods to solve a system of linear equations. Some of them are listed below:

  • Substitution method
  • Elimination method
  • Graphing method
  • Cramer’s rule for solving systems of linear equations

Let us learn about the different methods to solve a system of linear equations with examples.

Substitution Method

To understand the method of substitution, let us first consider an arbitrary system of linear equations with two variables:

ax + by = c              _ 1

dx + ey = f               _ 2

Where, a, b, c, d,e and f are constant terms.

Now, consider the equation 1,

x1
x2

Elimination Method

To understand the method of elimination, let us first consider an arbitrary system of linear equations with two variables:

ax + by = c              _ 4

dx + ey = f               _ 5

Where, a, b, c, d,e and f are constant terms.

Now, multiplying e with the equation 4 and multiplying b with the equation 5 to obtain the same coefficients of y for both the equations:

aex + bey = ce       _ 6

bdx + bey = bf       _ 7

Now, subtracting equation 6 from the equation 7:

bdx + bey – aex + bey = bf – ce

x3

Graphing Method

Graphing method is another method to solve a system of equations. For this, let us consider the system of equations with known coefficients:

2x + 3y = 6     _ 10

4x + 5y = 20     _ 11

Convert both equations in intercept form:

x4

Now, plot the given curves in the graph (Fig. 1):

x5

Cramer’s Rule for Solving Systems of Linear Equations

Let us consider three linear equations involving three variables:

a1x + b1y + c1z = d1  _ 14

a2x + b2y + c2z = d2  _ 15

a3x + b3y + c3z = d3  _ 16

Now, the coefficient matrix can be given by,

A=

x6
x7
x8
x9

There are some conditions under which Cramer’s Rule must be used. They are as follows:

  1. If the determinant of coefficient matrix, A is zero i.e. A = 0 and any one of the values of x, y and z is non-zero, then the system of equations has no solution.
  2. If the determinant of coefficient matrix, A as well as the values of x, y and z are zero i.e. A = x = y = z = 0, then the system of equations has an infinite number of solutions.
  3. If the determinant of coefficient matrix, A is non-zero i.e. A≠0, then the system of equation has a unique solution.

Conclusion

In this article, we learned about the different types of methods of solving linear equations. All these methods yield the same solutions. The first two methods, substitution and elimination methods, are more feasible and reliable than the graphing method to solve the system of linear equations. If there is a system of equations with more than two variables, then Cramer’s rule is preferable.

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