We are tasked with expressing the vector $v = \langle 1, 0, 5 \rangle$ as a linear combination of two given vectors $x = \langle 1, -4, 1 \rangle$ and $y = \langle 2, -4, 6 \rangle$.

A linear combination of vectors $x$ and $y$ takes the form:

$v = a \cdot x + b \cdot y$

This can be written as:

$\langle 1, 0, 5 \rangle = a \cdot \langle 1, -4, 1 \rangle + b \cdot \langle 2, -4, 6 \rangle$

Expanding both sides:

$\langle 1, 0, 5 \rangle = \langle a \cdot 1 + b \cdot 2, a \cdot (-4) + b \cdot (-4), a \cdot 1 + b \cdot 6 \rangle$

This gives us three equations corresponding to the components of the vectors:

1. $a + 2b = 1$
2. $-4a - 4b = 0$
3. $a + 6b = 5$

Now, let's solve this system of equations.

Step 1: Solve the second equation

From the second equation:

$-4a - 4b = 0 \quad \Rightarrow \quad a + b = 0 \quad \Rightarrow \quad a = -b$

Step 2: Substitute $a = -b$ into the first and third equations

Substitute into the first equation:

$(-b) + 2b = 1 \quad \Rightarrow \quad b = 1$

Substitute $b = 1$ into $a = -b$:

$a = -1$

Step 3: Verify with the third equation

Substitute $a = -1$ and $b = 1$ into the third equation:

$(-1) + 6(1) = 5 \quad \Rightarrow \quad -1 + 6 = 5$

The third equation is satisfied.

Thus, the solution is $a = -1$ and $b = 1$.

Final Answer:

The vector $v = \langle 1, 0, 5 \rangle$ can be written as:

$v = -1 \cdot x + 1 \cdot y$

Or more explicitly:

$v = - \langle 1, -4, 1 \rangle + \langle 2, -4, 6 \rangle$

Would you like more details on any step? Here are some related questions to consider:

1. What is the general method for solving systems of equations in linear algebra?
2. Can this method be extended to more than two vectors?
3. How do you check if two vectors are linearly independent?
4. What is the geometric interpretation of expressing a vector as a linear combination?
5. How would the process change if there were more than two vectors involved?

Tip: When solving systems of equations, substitution is often quicker when one equation can be simplified early on.