Solving Linear Equations

An equation in Mathematics is a Mathematical representation of real life problems as equivalent relationships in the form of variables and constants. Algebraic equations consist of algebraic expressions on either side of the ‘=’ sign. An equation consists of variables which are generally English alphabets that can take any real value for which the equation is satisfied. They also contain constants and coefficients. Constants are the variable independent numbers in the equation and the coefficients are the numbers associated with the variables. Linear equations are the algebraic equations with the order equal to unity or one. 

Types of Linear Equations:

Linear equations are the equations in which the variables are raised to the power equal to one. The linear equations are classified into different types based on the number of variables in the equation. The different types of linear equations are:

1. Linear equations in one variable are the linear equations which consist of only one variable

2. Linear equations in two variables are the linear equations which consist of two variables.

3. Linear equations in three variables are the linear equations which consist of three variables.

Standard form of Linear Equations:

Standard form of linear equation eases the method of solving the equation. The linear equation in one variable is written in standard form as ax + b = 0

The linear equation in two variable is written in standard form as ax + by + c = 0

The standard form of linear equation in three variable is ax + by + cz + d = 0

In the above equations, x, y and z are the variables. 

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Different Forms of Representing Linear Equations in two variables:

In the above equations, m is the slope, x and y are variables, A and B are the coefficients of x and y respectively. 

Solving Linear Equations in One Variable:

Finding the solution of a linear equation means finding the value of a the variable for which the equation becomes true. Linear equations generally have only one solution for each variable. For solving the linear equations in one variable, inverse Mathematical operations are used. 

Linear equations examples (one variable):

Solve 5x - 3/2 = 7

Solution:

5x - 3/2 = 7

5x = 7 + 3/2 

5x = 17/2

x = 17/10

Solving linear Equations in Two Variables:

A pair of linear equations in two variables can be solved using three different methods. They are:

• Elimination method

• Substitution method

• Cross multiplication method

In the elimination method, one variable is eliminated by subtracting or adding both the equations. In the substitution method, one variable is represented in the form of another variable in one of the equations. This value is substituted in the second equation to get the solution. In cross multiplication method, the solution of x and y is calculated using the formulae:

x =  \[\frac{b_{1} c_{2} - b_{2} c_{1}}{a_{1}b_{2} - a_{2}b_{1}}\] and y = \[\frac{c_{1} a_{2} - a_{1} b_{2}}{a_{2}b_{2} - a_{2}b_{1}}\]

Linear Equations Examples (2 variable):

Question : Solve x + y = 2 and x - y = 14 by all the three methods.

Solution: 

Case 1: Elimination method

x + y = 2 and x - y = 14 are added. If they are added, the variable ‘y’ can be eliminated.

=> x + y + x - y = 2 + 14 

=> 2x = 16

=> x = 16/2

=> x = 8

Now, substitute x = 7 in any one of the equations.

x - y = 14 => 8 - y = 14 

=> - y = 14 - 8 

=> y = - 6

The solution of the pair of equations is x = 7 and y = - 6.

Case 2: Substitution method

The first equation x + y = 2 => y = 2 - x

Substituting y = 2 - x in the second equation,

x - y = 14 => x - (2 - x) = 14 

=> x - 2 + x = 14

=> 2x = 14 + 2

=> 2x = 16

=> x = 16/2 

=> x = 8

Substituting x = 8 in first equation,

x + y = 2 => 8 + y = 2 => y = 2 - 8 => y = - 6

The solution of a pair of equations by substitution method is x = 8 and y = - 6.

Case 3: Cross Multiplication method

The pair of equations are x + y = 2 and x - y = 14

There two equations are written in standard form as: x + y - 2 = 0 and x - y - 14 = 0

So, a1 = 1, b1 = 1 and c1 = -2; a2 = 1, b2 = -1 and c2 = - 14

Using the formula,

X = \[\frac{b_{1} c_{2} - b_{2}c_{1}}{a_{1}b_{2} - a_{2}b_{1}}\] and y = \[\frac{c_{1} a_{2} - c_{2}a_{1}}{a_{1}b_{2} - a_{2}b_{1}}\]

x =   \[\frac{(1 × - 14) - (-1 × - 2)}{(1× -1) - (1 × 1)}\] and y =  \[\frac{(-2 × - 1) - (-14 × - 1)}{(1× -1) - (1 × 1)}\]

x  = \[\frac{-14 -2}{-1-1}\] and y = \[\frac{-2-(-14)}{-1-1}\]

x  =  \[\frac{-16}{-2}\] and y = \[\frac{12}{-2}\]

x = 8 and y = - 6

Fun Facts:

Basic Mathematical operations and their inverse operations are listed in the table below.

• The solution of a pair of linear equations examples in two variables remains the same irrespective of the method used to find to solve the equations.

• Linear equations in two variables can also be solved using determinant methods.