Determine if the system has a nontrivial solution. Try to use as few row operations as possible. 6x^2 + 7xz 6x^3 + 4x^3 Choose the correct answer below: The system has only a trivial solution: The system has a nontrivial solution: It is impossible to determine.
The Correct Answer and Explanation is:
ChatGPT said:
To determine whether a system has a nontrivial solution, we need to understand what the system is. However, your question seems to mix a few different expressions:
“6x² + 7xz 6x³ + 4x³” — this looks more like a list of algebraic expressions than a system of equations. Let’s interpret it carefully.
Step 1: Clarify the Problem
You’re likely being asked to determine if a system of equations (possibly homogeneous) has a nontrivial solution. A homogeneous system has the form:Ax⃗=0⃗A \vec{x} = \vec{0}Ax=0
and has a nontrivial solution (i.e., a solution other than the zero vector) if and only if the system has free variables, or equivalently, if the determinant of the coefficient matrix is zero (i.e., the matrix is not full rank).
But from what you typed:
- “6x² + 7xz 6x³ + 4x³” could be meant to represent the expressions in equations, or possibly a vector form.
Let’s assume this is a homogeneous system where the coefficients or expressions might represent rows of a matrix.
Suppose we are given a vector space expression like:v⃗1=6×2+7xz,v⃗2=6×3+4×3=10×3\vec{v}_1 = 6x^2 + 7xz, \quad \vec{v}_2 = 6x^3 + 4x^3 = 10x^3v1=6×2+7xz,v2=6×3+4×3=10×3
If these vectors are viewed as linear combinations or parts of a homogeneous system, we ask:
Do these expressions (vectors) form a linearly independent set?
Let’s consider the following:
- The expression 6×2+7xz6x^2 + 7xz6x2+7xz involves degree-2 monomials: x2x^2×2, xzxzxz
- The expression 10x310x^310×3 involves only the degree-3 monomial x3x^3×3
Since these expressions involve different degrees and distinct monomials, they are linearly independent.
Thus, if these are rows or columns of a matrix representing a homogeneous system, and if the number of unknowns exceeds the number of independent equations, there would be a nontrivial solution.
✅ Final Answer:
The system has a nontrivial solution.
Explanation
A homogeneous system of equations is one in which all equations are set equal to zero, such as:a1x+a2y+a3z=0a_1x + a_2y + a_3z = 0a1x+a2y+a3z=0
A trivial solution occurs when all variables are zero (e.g., x=y=z=0x = y = z = 0x=y=z=0). A nontrivial solution exists if there are nonzero values of the variables that also satisfy the system.
To determine whether a nontrivial solution exists, we analyze the linear independence of the expressions provided.
Given the expressions:6×2+7xz,6×3+4×3=10x36x^2 + 7xz,\quad 6x^3 + 4x^3 = 10x^36×2+7xz,6×3+4×3=10×3
These can be seen as vectors in a polynomial space. The first expression, 6×2+7xz6x^2 + 7xz6x2+7xz, contains monomials x2x^2×2 and xzxzxz, both of degree 2. The second expression, 10x310x^310×3, contains only x3x^3×3, which is of degree 3.
Because these expressions involve distinct monomials and are of different degrees, they cannot be linear combinations of one another. Therefore, they are linearly independent.
If such expressions form part of a matrix in a homogeneous linear system, then each independent vector adds to the rank of the matrix. A system has a nontrivial solution if the number of variables exceeds the rank (i.e., the number of linearly independent equations). Since these expressions do not overlap in their monomial components, and assuming there are at least 3 variables (e.g., x,y,zx, y, zx,y,z), we would have more variables than independent equations.
Thus, the system allows for at least one free variable, which implies the existence of nontrivial solutions.
Hence, the correct conclusion is:
✅ The system has a nontrivial solution
