Which classification describes the following system of equations?
$\left\{\begin{array}{c}
5 x+y+2 z=13 \\
4 x+3 y+6 z=17 \\
x+2 y=4
\end{array}\right.$
A. inconsistent and dependent
B. consistent and dependent
C. consistent and independent
D. inconsistent and independent
Answer (1)
Calculate the determinant of the coefficient matrix: det ( A ) = − 44 .
Since the determinant is non-zero, the system is consistent and independent.
A consistent and independent system has a unique solution.
The system of equations is classified as co n s i s t e n t an d in d e p e n d e n t .
Explanation
Understanding the Problem We are given a system of three linear equations with three unknowns: x , y , and z . The equations are:
{ 5 x + y + 2 z = 13 4 x + 3 y + 6 z = 17 x + 2 y = 4
Our goal is to classify this system as either consistent or inconsistent, and either dependent or independent.
Setting up the Coefficient Matrix To classify the system, we can first find the determinant of the coefficient matrix. The coefficient matrix A is formed by the coefficients of x , y , and z in the equations:
A = [ 5 1 2 4 3 6 1 2 0 ]
We calculate the determinant of A .
Calculating the Determinant The determinant of A is calculated as follows:
det ( A ) = 5 ( 3 ⋅ 0 − 6 ⋅ 2 ) − 1 ( 4 ⋅ 0 − 6 ⋅ 1 ) + 2 ( 4 ⋅ 2 − 3 ⋅ 1 ) = 5 ( − 12 ) − 1 ( − 6 ) + 2 ( 8 − 3 ) = − 60 + 6 + 10 = − 44
Since the determinant is non-zero ( det ( A ) = − 44 = 0 ), the system has a unique solution.
Determining Consistency and Independence Since the determinant of the coefficient matrix is non-zero, the system of equations is consistent and independent. This means that there is exactly one solution for x , y , and z that satisfies all three equations.
Final Classification Therefore, the classification of the system of equations is consistent and independent.
Examples
Consider a scenario where you need to determine the exact amounts of three ingredients to create a specific mixture. Each ingredient contributes differently to the final product, and you have three equations that relate the amounts of these ingredients to the desired properties of the mixture. If the system of equations is consistent and independent, it means there is a unique solution, and you can find the exact amounts of each ingredient needed to achieve the desired outcome. This is applicable in fields like chemistry, cooking, and manufacturing where precise proportions are crucial.
