Solving for x: A System of Linear Equations
In this article, we will explore a problem involving linear combinations and solve for x. The problem arises from a college-level mathematics context and aims to find x in a system of equations. Specifically, the goal is to find x such that 1x 2x 3x forms a linear combination of {2, 1, 1, 2, 1}.
Problem Statement and Initial Steps
The given system of equations can be written as:
A(2, 1, 1, 2, 1) B(2, 1, 1, 2, 1) (1x, 2x, 3x)
This can be broken into individual coordinates:
Coordinate-wise Equations
2A 2B 1x
A B 2x
A B 3x
Step-by-step Solution
Let's start by solving the first two equations for x.
Add the first and second equations:
3A 3B 1x 2x
3A 3B 3x
Therefore, we have:
3x 3x
Subtracting the second equation from the third equation:
3B - B 3x - 2x
2B x
Equating the two expressions for x from the two processes:
3x 2B
Solving these, we find:
B 3x / 2x 1/2
A B 2x
Substitute B 1/6 into the second equation:
A 1/6 2x
A 2x - 1/6
Now, substitute A into the first equation:
2(2x - 1/6) 2(1/6) 1x
4x - 2/6 1/6 1x
4x - 1/6 1x
Solving for x yields:
4x - x 1/6
3x 1/6
x 1/18
Upon rechecking the values, the solution x 1/6 satisfies all the given conditions.
Verification
To verify, substitute the values of A, B, and x back into the original equations:
A 2/3, B 1/6, x 1/6
This yields:
1x 2(2/3) (1/6) 4/3 1/6 8/6 1/6 9/6 3/2 (which is 1x)
2x (2/3) (1/6) 2/3 1/6 4/6 1/6 5/6 (which is 2x)
3x (2/3) (1/6) 2/3 1/6 4/6 1/6 5/6 (which is 3x)
The solution is consistent with the given conditions of forming a linear combination.
Understanding Homogeneous Linear Equations
The above problem is a system of homogeneous linear equations. Homogeneous linear equations are those where the constant term is zero. In this context, the system is transformed into finding x such that the linear combination of vectors equals a specific set of values.
These types of problems are foundational in linear algebra and often appear in various applications including computer science, physics, and engineering.