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Consider the following set of equations:
x + 2y = 5
4x + 8y = 12
3x + 6y + 3z = 15
This set
x + 2y = 5
4x + 8y = 12
3x + 6y + 3z = 15
This set
- a)Has a unique solution
- b)Has no solution
- c)Has finite number of solutions
- d)Has infinite number of solutions
Correct answer is option 'B'. Can you explain this answer?
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Verified Answer
Consider the following set of equations:x + 2y = 54x + 8y = 123x + 6y ...
Set of equations is
Above set of equations can be written as
Augmented matrix [AB] is given as
Performing gauss-Elimination on the above matrix
Above set of equations can be written as
Augmented matrix [AB] is given as
Performing gauss-Elimination on the above matrix
Most Upvoted Answer
Consider the following set of equations:x + 2y = 54x + 8y = 123x + 6y ...
Given Equations:
1) x + 2y = 54
2) x + 8y = 123
3) x + 6y + 3z = 15
Explanation:
To determine the number of solutions for the given set of equations, we can use the concept of simultaneous equations and matrix algebra.
Step 1: Write the equations in matrix form:
Let's write the given equations in matrix form as AX = B, where A is the coefficient matrix, X is the variable matrix, and B is the constant matrix.
The coefficient matrix A can be written as:
A = | 1 2 0 |
| 1 8 0 |
| 1 6 3 |
The variable matrix X can be written as:
X = | x |
| y |
| z |
The constant matrix B can be written as:
B = | 54 |
| 123 |
| 15 |
Step 2: Determine the determinant of the coefficient matrix A:
If the determinant of matrix A is non-zero, then the system of equations has a unique solution. Otherwise, it has either no solution or infinite solutions.
Calculating the determinant of A, we have:
|A| = | 1 2 0 |
| 1 8 0 |
| 1 6 3 |
Expanding along the first row, we get:
|A| = 1 * | 8 0 |
-1 * | 6 3 |
Calculating the determinants of the 2x2 matrices, we have:
|A| = 1 * (8 * 3 - 0 * 6) - (-1) * (6 * 3 - 0 * 8)
= 1 * 24 - (-1) * 18
= 24 + 18
= 42
Step 3: Determine the number of solutions:
Since the determinant of A is non-zero (|A| ≠ 0), the given set of equations has a unique solution.
Therefore, the correct answer is option 'A' - Has a unique solution.
1) x + 2y = 54
2) x + 8y = 123
3) x + 6y + 3z = 15
Explanation:
To determine the number of solutions for the given set of equations, we can use the concept of simultaneous equations and matrix algebra.
Step 1: Write the equations in matrix form:
Let's write the given equations in matrix form as AX = B, where A is the coefficient matrix, X is the variable matrix, and B is the constant matrix.
The coefficient matrix A can be written as:
A = | 1 2 0 |
| 1 8 0 |
| 1 6 3 |
The variable matrix X can be written as:
X = | x |
| y |
| z |
The constant matrix B can be written as:
B = | 54 |
| 123 |
| 15 |
Step 2: Determine the determinant of the coefficient matrix A:
If the determinant of matrix A is non-zero, then the system of equations has a unique solution. Otherwise, it has either no solution or infinite solutions.
Calculating the determinant of A, we have:
|A| = | 1 2 0 |
| 1 8 0 |
| 1 6 3 |
Expanding along the first row, we get:
|A| = 1 * | 8 0 |
-1 * | 6 3 |
Calculating the determinants of the 2x2 matrices, we have:
|A| = 1 * (8 * 3 - 0 * 6) - (-1) * (6 * 3 - 0 * 8)
= 1 * 24 - (-1) * 18
= 24 + 18
= 42
Step 3: Determine the number of solutions:
Since the determinant of A is non-zero (|A| ≠ 0), the given set of equations has a unique solution.
Therefore, the correct answer is option 'A' - Has a unique solution.
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Consider the following set of equations:x + 2y = 54x + 8y = 123x + 6y + 3z = 15This seta)Has a unique solutionb)Has no solutionc)Has finite number of solutionsd)Has infinite number of solutionsCorrect answer is option 'B'. Can you explain this answer?
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