The equations x + 2y + 3z = 1 2x + y + 3z = 2
have a unique solution
Solving Systems of Linear Equations: Determining the Nature of Solutions
We are given a system of three linear equations with three variables x, y, and z:
- \(x + 2y + 3z = 12\)
- \(x + y + 3z = 25\)
- \(x + 5y + 9z = 4\)
To determine whether this system of linear equations has a unique solution, infinitely many solutions, or is inconsistent, we can use the concept of determinants from matrix theory.
Matrix Representation of the System
The given system can be written in the matrix form \(AX = B\), where:
A is the coefficient matrix:
| 1 | 2 | 3 |
| 1 | 1 | 3 |
| 1 | 5 | 9 |
X is the variable matrix:
| x |
| y |
| z |
B is the constant matrix:
| 12 |
| 25 |
| 4 |
Using the Determinant of the Coefficient Matrix
The nature of the solution of a system of linear equations \(AX = B\) depends on the determinant of the coefficient matrix A, denoted as \(\det(A)\) or \(|A|\).
- If \(\det(A) \neq 0\), the matrix A is non-singular, and the system has a unique solution.
- If \(\det(A) = 0\), the matrix A is singular. In this case, the system may have infinitely many solutions or no solution (be inconsistent). To distinguish between these two cases, we would typically look at the augmented matrix \([A|B]\) and its rank, or examine the determinants of matrices formed by replacing columns of A with B (Cramer's Rule numerators).
Calculating the Determinant of A
Let's calculate the determinant of the coefficient matrix A:
$$ A = \begin{pmatrix} 1 & 2 & 3 \\ 1 & 1 & 3 \\ 1 & 5 & 9 \end{pmatrix} $$We calculate the determinant by expanding along the first row:
$$ \det(A) = 1 \cdot \begin{vmatrix} 1 & 3 \\ 5 & 9 \end{vmatrix} - 2 \cdot \begin{vmatrix} 1 & 3 \\ 1 & 9 \end{vmatrix} + 3 \cdot \begin{vmatrix} 1 & 1 \\ 1 & 5 \end{vmatrix} $$ $$ \det(A) = 1 \cdot ((1 \times 9) - (3 \times 5)) - 2 \cdot ((1 \times 9) - (3 \times 1)) + 3 \cdot ((1 \times 5) - (1 \times 1)) $$ $$ \det(A) = 1 \cdot (9 - 15) - 2 \cdot (9 - 3) + 3 \cdot (5 - 1) $$ $$ \det(A) = 1 \cdot (-6) - 2 \cdot (6) + 3 \cdot (4) $$ $$ \det(A) = -6 - 12 + 12 $$ $$ \det(A) = -6 $$Conclusion on the Nature of Solution
We found that the determinant of the coefficient matrix A is \(\det(A) = -6\). Since \(\det(A) \neq 0\), the matrix A is non-singular. According to the rules for the nature of solutions of linear systems, a non-singular coefficient matrix indicates that the system of linear equations has a unique solution.
Therefore, the given system of equations \(x + 2y + 3z = 12\), \(x + y + 3z = 25\), and \(x + 5y + 9z = 4\) has a unique solution.
Revision Table: System of Linear Equations
| Condition on \(\det(A)\) | Nature of Solutions | Method to Find Solution |
|---|---|---|
| \(\det(A) \neq 0\) | Unique Solution | Matrix Inversion (\(X = A^{-1}B\)), Cramer's Rule, Gaussian Elimination |
| \(\det(A) = 0\) and \(\text{rank}(A) = \text{rank}([A|B])\) < number of variables | Infinitely Many Solutions | Gaussian Elimination to parametric form |
| \(\det(A) = 0\) and \(\text{rank}(A) \neq \text{rank}([A|B])\) | No Solution (Inconsistent) | Check ranks of A and augmented matrix [A|B] |
Additional Information: Methods for Solving Linear Systems
Once the nature of the solution is determined, if a unique solution exists, various methods can be used to find the values of x, y, and z:
- Cramer's Rule: This method uses determinants of matrices formed by replacing columns of A with the constant vector B. It is generally efficient for systems with a unique solution, especially smaller ones.
- Matrix Inversion Method: If \(\det(A) \neq 0\), the inverse matrix \(A^{-1}\) exists. The solution is then given by \(X = A^{-1}B\). This involves calculating the inverse of A and performing matrix multiplication.
- Gaussian Elimination (Row Reduction): This method involves performing elementary row operations on the augmented matrix \([A|B]\) to transform it into row echelon form or reduced row echelon form. This method is versatile and can determine consistency, the number of solutions, and the solution itself.
- Substitution or Elimination: These are algebraic methods where variables are eliminated or substituted iteratively across the equations until the values of all variables are found.
For the given system, since a unique solution exists, any of these methods could be applied to find the specific values of x, y, and z.
Similar Questions
The system of equations
2x + y - 3z = 5
3x - 2y + 2z = 5 and
5x - 3y - z = 16The system of equations kx + y + z = 1, x + ky + z = k and x + y + kz = k 2has no solution if k equals
Consider the determinant
Δ = \(\left|\begin{array}{lll}\text{a}_{11} & \text{a}_{12} & \text{a}_{13} \\ \text{a}_{21} & \text{a}_{22} & \text{a}_{23} \\ \text{a}_{31} & \text{a}_{32} & \text{a}_{33}\end{array}\right|\)
If a 13 = yz, a 23 = zx, a 33 = xy and the minors of a 13 , a 23 , a 33 are respectively (z − y), (z − x), (y − x) then what is the value of Δ ?
What is the value of a 11 C11 + a 12 C12 + a 13 C13 ?
What is the value of a 21 C11 + a 22 C12 + a 23 C13 ?
What is the value of \(\left|\begin{array}{lll}\text{a}_{21} & \text{a}_{31} & \text{a}_{11} \\ \text{a}_{23} & \text{a}_{33} & \text{a}_{13} \\ \text{a}_{22} & \text{a}_{32} & \text{a}_{12}\end{array}\right|\) ?
The system of linear equation kx + y + z = 1, x + ky + z = 1 and x + y + kz = 1 has a unique solution under which one of the following conditions?
Consider the set A of all matrices of order 3 × 3 with entries 0 or 1 only. Let B be the subset of A consisting of all matrices whose determinant is 1. Let C be the subset of A consisting of all matrices whose determinant is -1. Then which one of the following is correct?
For what values of k is the system of equations 2k 2x + 3y - 1 = 0, 7x - 2y + 3 = 0, 6kx + y + 1 = 0 consistent?
Which of the following are correct in respect of the system of equation
x + y + z = 8,
x – y + 2z = 6 and
3x – y + 5z = k?
1. They have no solution if k = 15
2. They have infinitely many solutions, if k = 20
3. They have a unique solution if k = 25
Select the correct answer using the code given below:
Important Questions from Application of Determinants
The equations 3x - 4y = 5 and 12x - 16y = 20 have:
The system of equations
2x + y - 3z = 5
3x - 2y + 2z = 5 and
5x - 3y - z = 16The system of equations kx + y + z = 1, x + ky + z = k and x + y + kz = k 2has no solution if k equals
The equations 2x - ky + 7 = 0 and 6x - 12y + 15 = 0 have no solution for
If (a, b), (x1, y1) and (x2, y2) are the vertices of a triangle such that the x-coordinates a, x1, x2 are in geometric progression with common ratio r and the y-coordinates b, y1, y2 are also in geometric progression with common ratio s, then the area of the triangle is:
