Question 27 of 58intermediate🔧 ApplyLong Answer4 marks

Examine the consistency of the system of equations:

2x−y=52x - y = 5

x+y=4x + y = 4

Correct Answer

The system of equations is consistent.

Exercise: EXERCISE 4.5 | Q: 2 | (Chapter: 22)
For More Understanding

Explanation

This question is from Exercise 4.5, Question 2 of the NCERT textbook. To examine consistency, we express the system in matrix form AX=BAX = B and check if the coefficient matrix AA is nonsingular.

If ∣A∣≠0|A| \neq 0, the system has a unique solution and is consistent. If ∣A∣=0|A| = 0, further investigation is needed using (adj A)B(\text{adj } A)B.

Solution Steps

  1. Step 1: Write the system in matrix form AX=BAX = B, where A=[2−111]A = \begin{bmatrix} 2 & -1 \\ 1 & 1 \end{bmatrix}, X=[xy]X = \begin{bmatrix} x \\ y \end{bmatrix}, and B=[54]B = \begin{bmatrix} 5 \\ 4 \end{bmatrix}.

  2. Step 2: Calculate the determinant of matrix AA:

    ∣A∣=2(1)−(−1)(1)=2+1=3|A| = 2(1) - (-1)(1) = 2 + 1 = \mathbf{3}

  3. Step 3: Since ∣A∣=3≠0|A| = 3 \neq 0, the matrix AA is nonsingular.

  4. Step 4: A nonsingular matrix has an inverse, so the system has a unique solution.

  5. Step 5: Therefore, the system of equations is consistent.

PreviousNext