# System of Equations

Set of two or more equations which contain same variables is known as a system of equations.

The equations a_{1 }x + b_{1 }y + c_{1} = 0 and a_{2 }x + b_{2 }y + c_{2} = 0 make a system of equations.

# Methods of solving a system of simultaneous linear equations

**Method of substitution**Find the value of y in terms of the other variable x from any one of the two equations.**Step-1.**Substitute this value of y obtained in Step-1 in the other equation. Then, solve this equation to get the value of x.**Step-2.**Substitute the value of x obtained in Step-2 in any of the given equations and find the value of y.**Step-3.**

ExampleSolve for*x*and*y*:

Solution

Letâ€™s first find x in terms of y from equation (1)

Substitute the value of x in (2)

Substitute the value of y in (1)

**Method of Elimination****Step-1.**Find the L C M of coefficients of one variable (say x) in the given equations. Multiply the equations by a suitable number such that coefficients of x are numerically equal (to the L C M ) in both the equations.**Step-2.****Step-3.**Solve the equation obtained in Step-2 as the linear equation of one variable y.**Step-4.**Substitute the value of y obtained in the above step in any of the equations to obtain the value of the other variable (x).ExampleSolveSolution**Let****â€¦â€¦â€¦â€¦â€¦**(1)**â€¦â€¦â€¦â€¦â€¦**(2)**â€¦â€¦â€¦â€¦â€¦**(3)**â€¦â€¦â€¦â€¦â€¦**(4)**Method of cross-multiplication****Step-1.**Write down the given system of equations as shown below. Bring all the terms on one side leaving the right hand side to be zero._{1 }x + b_{1 }y + c_{1}= 0_{2 }x + b_{2 }y + c_{2}= 0**Step-2.**Write the coefficients, starting with the coefficients of y, then the constant terms, followed by the coefficients of x, and repeat the coefficients of y as shown below:**Step-3.**Write the coefficients of x, y and 1 in the following manner:

ExampleSolveSolution

Eq. (1) and Eq. (2) can be written as

**There are two methods of finding the solution of a system of linear equation in three variables they are:**

# Method of solving simultaneous linear equation with 3 variables

- Method of elimination
- Method of cross-multiplication

The methods are similar to the one already discussed for the solution system of linear equation in two variables.

Example

**Solve**for

*x*,

*y*and

*z*:

Solution

**(a) Method of elimination**

2x â€“ y + z = 3â€¦â€¦â€¦â€¦â€¦â€¦â€¦.(1)

x + 3y â€“ 2z = 11â€¦â€¦â€¦â€¦â€¦â€¦â€¦. (2)

3x â€“ 2y + 4z = 1â€¦â€¦â€¦â€¦â€¦â€¦â€¦. (3)

By Eq. (1) Ã— 2 we get

4x â€“ 2y + 2z = 6â€¦â€¦â€¦â€¦â€¦â€¦â€¦. (4)

By Eq. (2) + Eq. (4), 5x + y = 17â€¦â€¦(5) [the variable z is thus eliminated]

By Eq. (2) Ã— 2, 2x + 6y â€“ 4z = 22â€¦â€¦â€¦â€¦â€¦â€¦â€¦. (6)

By Eq. (3) + Eq. (6), 5x + 4y = 23â€¦â€¦â€¦â€¦â€¦â€¦â€¦. (7)

By Eq. (5) â€“ Eq. (7), â€“3y = â€“ 6 or y = 2

Putting y = 2 in Eq. (5), 5x + 2 = 17, or 5x = 15 or, x = 3

Putting x = 3 and y = 2 in Eq. (1)

2 Ã— 3 â€“ 2 + z = 3

or 6 â€“ 2 + z = 3

or 4 + z = 3

or z = â€“1

So, x = 3, y = 2, z = â€“1 is the required solution.

(Any two of 3 equations can be chosen for elimination of one of the variables.)

**(b) Method of cross-multiplication**

We write the equations as follows:

2x â€“ y + ( z â€“ 3) = 0

x + 3y + (â€“2 z â€“11) = 0

By cross-multiplication,

Substituting above values for x and y in Eq. (3), i.e., 3x â€“ 2y + 4z = 1, we have

or 60 â€“ 3z â€“ 10z â€“ 38 + 28z = 7

or 15z = 7 â€“ 22 or 15z = â€“15 or z = â€“1

Now

Thus, x = 3, y = 2, z = â€“1

Example

Two numbers are such that sum of five times the first and thrice the second is 7. Also, the difference of three times the first and twice the second is 8. Find the numbers.

Solution

Both the numbers are unknown. Letâ€™s assume the first number as

As given in the question, the first condition says,

5x + 3y = 7â€¦â€¦â€¦â€¦â€¦â€¦â€¦(1)

The second condition says,

3x â€“ 2y = 8â€¦â€¦â€¦â€¦â€¦â€¦â€¦ (2)

Now, we have two linear equations in two variables. We can apply any of the 3 methods to solve these equations. Let us use elimination method here.

Coefficients of x in the equations are 5 and 3. L C M of 5 and 3 is 15.

We can multiply the first equation by 3 and second equation by 5 to make both the coefficients equal to 15.

â€¦â€¦â€¦â€¦â€¦â€¦â€¦ (3)

â€¦â€¦â€¦â€¦â€¦â€¦â€¦ (4)

Now, coefficients of x in Eqs. (3) and (4) are equal. So, we can subtract Eq. 4 from Eq. 3 to eliminate x and find y.

We can find x by substituting this value of y in any of the given equations. Substituting in Eq. (1) we get,

*x*and the second numbers as*y*.As given in the question, the first condition says,

5x + 3y = 7â€¦â€¦â€¦â€¦â€¦â€¦â€¦(1)

The second condition says,

3x â€“ 2y = 8â€¦â€¦â€¦â€¦â€¦â€¦â€¦ (2)

Now, we have two linear equations in two variables. We can apply any of the 3 methods to solve these equations. Let us use elimination method here.

Coefficients of x in the equations are 5 and 3. L C M of 5 and 3 is 15.

We can multiply the first equation by 3 and second equation by 5 to make both the coefficients equal to 15.

â€¦â€¦â€¦â€¦â€¦â€¦â€¦ (3)

â€¦â€¦â€¦â€¦â€¦â€¦â€¦ (4)

Now, coefficients of x in Eqs. (3) and (4) are equal. So, we can subtract Eq. 4 from Eq. 3 to eliminate x and find y.

We can find x by substituting this value of y in any of the given equations. Substituting in Eq. (1) we get,

Example

In a fraction, if 1 is added to the denominator, the fraction becomes 1/2. If 2 is added to the numerator, the fraction becomes 1. Find the fraction.

Solution

Both numerator and denominator of the fraction are unknown. Letâ€™s assume numerator to be

The first condition can be written as .

This gives us

The second condition can be written as

Now, we have a system of linear equations which can be solved using any of the three methods.

Coefficient of y in both equations is â€“1. We can directly subtract Eq. (2) from Eq. (1) to eliminate y.

Substituting this value of x in Eq. (2), we get,

Hence, the required fraction becomes .

*x*and denominator to be*y*. The fraction becomes .The first condition can be written as .

This gives us

The second condition can be written as

Now, we have a system of linear equations which can be solved using any of the three methods.

Coefficient of y in both equations is â€“1. We can directly subtract Eq. (2) from Eq. (1) to eliminate y.

Substituting this value of x in Eq. (2), we get,

Hence, the required fraction becomes .

**A system having two simultaneous linear equations is said to be consistent if it has at least one solution (i.e., it has one or more than one solution).**

# Consistent and inconsistent systems of two simultaneous linear equations

A system having two simultaneous linear equations is said to be inconsistent if it has no solution.

Consider two simultaneous linear equations, say

Hence, it is consistent and the graph is a pair of intersecting lines.

Hence, it is consistent and the graph is a pair of overlapping lines.

Hence, it is inconsistent and the graph is a pair of parallel lines.

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