Given below is a system of linear equations.5a + b + 3c = 59a + c = -8a + 2b + 5c=8This system of equations has,a)A unique solutionb)Infinitely many solutionc)No solutiond)Cannot be determinedCorrect answer is option 'B'. Can you explain this answer?

Question

Accepted Answer

Comparing the system of equation with the matrix equation,

A.X = B

We have,

A =

, B =

Augmented matrix is given by,

Applying, R₃→R₃−(2R₁−R₂)

We get,

Thus, Rank(A)=Rank(A:B)=2 < />

Hence, the system of equations is consistent and has infinitely many solutions.

Answer

Given system of linear equations:
5a + b + 3c = 5
a + c = -8
a + 2b + 5c = 8

Explanation:
To determine the nature of the system of linear equations, we can use the concept of determinant. The determinant can be calculated by forming a coefficient matrix and evaluating its determinant.

Step 1: Form the coefficient matrix:
The coefficient matrix is obtained by writing down the coefficients of the variables in the system of equations. In this case, the coefficient matrix is:
[5 1 3]
[1 0 1]
[1 2 5]

Step 2: Calculate the determinant:
The determinant of the coefficient matrix can be calculated using various methods, such as expansion by minors or using the properties of determinants. Here, we will use the expansion by minors method.
The determinant of the coefficient matrix is given by:
det = (5 * 0 * 5) + (1 * 1 * 1) + (3 * 1 * 2) - (3 * 0 * 1) - (5 * 1 * 1) - (2 * 1 * 5)
= 0 + 1 + 6 - 0 - 5 - 10
= 2

Step 3: Analyze the determinant:
If the determinant of the coefficient matrix is non-zero, then the system of linear equations has a unique solution. If the determinant is zero, then the system may have infinitely many solutions or no solution.

In this case, the determinant is 2, which is non-zero. Therefore, the given system of linear equations has a unique solution.

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