Question 34 of 58intermediate🔧 ApplyLong Answer4 marks

Examine the consistency of the system of equations: x + 2y = 2 2x + 3y = 3

Correct Answer

The system of equations is consistent.

Exercise: EXERCISE 4.5 | Q: 1 | (Page 22)
For More Understanding

Explanation

To examine the consistency of the system, we express the equations in the matrix form AX=BAX = B. We then calculate the determinant of the coefficient matrix AA. As demonstrated in Example 16 of the provided context, if the determinant ∣A∣≠0|A| \neq 0, the matrix is nonsingular, and the system has a unique solution, which implies the system is consistent. Since the calculated determinant is non-zero, the given system is consistent.

Solution Steps

  1. Step 1: Write the system of equations in matrix form AX=BAX = B. The given system is:

    x+2y=2x + 2y = 2 2x+3y=32x + 3y = 3

    This can be written as:

    A=[1223],X=[xy] and B=[23]A = \begin{bmatrix} 1 & 2 \\ 2 & 3 \end{bmatrix}, X = \begin{bmatrix} x \\ y \end{bmatrix} \text{ and } B = \begin{bmatrix} 2 \\ 3 \end{bmatrix}

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

    ∣A∣=1(3)−2(2)=3−4=−1|A| = 1(3) - 2(2) = 3 - 4 = -1

  3. Step 3: Analyze the determinant for consistency. Since ∣A∣=−1≠0|A| = -1 \neq 0, matrix AA is nonsingular. Therefore, A−1A^{-1} exists.

  4. Step 4: Conclude the consistency. Since AA is nonsingular, the system of equations has a unique solution. Hence, the system of equations is consistent.

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