Suppose v, w are in V. Explain why there is exists a unique x in V such that v+3x=w

To explain why there exists a unique \( x \) in \( V \) such that \( v + 3x = w \) for given vectors \( v \) and \( w \) in a vector space \( V \), we can use the properties of vector spaces, specifically the existence of additive inverses and scalar multiplication.

### Step 1: Rearrange the equation
We start with the equation \( v + 3x = w \). To isolate \( x \), we subtract \( v \) from both sides of the equation:
\[
3x = w - v
\]
Here, \( w - v \) is another vector in \( V \), obtained by adding the additive inverse of \( v \) to \( w \).

### Step 2: Use scalar multiplication
Next, we need to solve for \( x \). The equation \( 3x = w - v \) involves scalar multiplication. Since \( V \) is a vector space, scalar multiplication by \( \frac{1}{3} \) (which is possible because \( 3 \) is nonzero) is defined. We multiply both sides of the equation by \( \frac{1}{3} \) to solve for \( x \):
\[
x = \frac{1}{3}(w - v)
\]

### Step 3: Uniqueness of \( x \)
The solution \( x = \frac{1}{3}(w - v) \) is unique because the operations we performed—subtraction of vectors and multiplication by a scalar—are both well-defined and unique in a vector space. There is no other vector \( x \) in \( V \) that can satisfy the equation \( v + 3x = w \) because we derived \( x \) explicitly.

### Conclusion
Therefore, for any \( v, w \in V \), there exists a unique \( x \in V \) such that \( v + 3x = w \). The solution is given by \( x = \frac{1}{3}(w - v) \), and this solution is unique due to the properties of vector spaces.

Would you like further details or have any questions? Here are some related problems to consider:

1. Prove that if \( a \) is a nonzero scalar and \( v \) is a vector in \( V \), then there exists a unique \( x \) such that \( ax = v \).
2. Show that the equation \( v + ax = w \) has a unique solution for any \( a \neq 0 \) in the field and vectors \( v, w \) in \( V \).
3. Prove that the equation \( v + ax = w \) has no solutions if \( a = 0 \) unless \( v = w \).
4. Given two vectors \( v, w \) in \( V \), explain why there is a unique vector \( x \) such that \( v + x = w + x \).
5. Show that for any vector \( v \) in \( V \), the equation \( x + x + x = v \) has a unique solution.

**Tip:** The uniqueness of solutions in vector spaces often hinges on the non-zero scalar property. Always check if the scalar involved in multiplication is non-zero when solving linear equations.

Understand why there exists a unique solution x in vector space V such that v + 3x = w. This detailed explanation covers vector space properties, scalar multiplication, and the uniqueness of solutions. Ideal for advanced undergraduate students studying linear algebra.

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