Answer :
Answer:
True. A unique solution means there is only one solution to a problem or question. It is true because a unique solution implies that there is only one correct answer or way to solve a particular issue.
Answer:
Step-by-step explanation:To determine whether a mathematical problem or system has a unique solution, we need to consider the context of the problem. A unique solution implies that there is exactly one solution to the problem or system. The truth of this statement can vary depending on the type of mathematical problem we are dealing with. Let’s explore a few scenarios:
### 1. Linear Equations
For a system of linear equations, a unique solution exists if the system is consistent and the number of equations is equal to the number of variables, with the determinant of the coefficient matrix being non-zero.
**Example:**
Consider the system of linear equations:
\[
\begin{cases}
2x + 3y = 5 \\
x - y = 1
\end{cases}
\]
This system has a unique solution because the determinant of the coefficient matrix \(\begin{bmatrix} 2 & 3 \\ 1 & -1 \end{bmatrix}\) is non-zero (Determinant = 2*(-1) - 3*1 = -2 - 3 = -5 ≠ 0).
### 2. Quadratic Equations
For a quadratic equation \(ax^2 + bx + c = 0\), there can be zero, one, or two real solutions depending on the discriminant \(\Delta = b^2 - 4ac\):
- If \(\Delta > 0\), there are two distinct real solutions.
- If \(\Delta = 0\), there is exactly one real solution (a unique solution).
- If \(\Delta < 0\), there are no real solutions (but two complex solutions).
**Example:**
The quadratic equation \(x^2 - 4x + 4 = 0\) has a unique solution because the discriminant is zero (\(\Delta = 4^2 - 4*1*4 = 16 - 16 = 0\)).
### 3. Nonlinear Equations
For nonlinear equations, such as systems involving polynomials or transcendental functions, determining the uniqueness of solutions can be more complex and often requires specific methods of analysis.
**Example:**
The equation \(e^x = x + 2\) can be analyzed graphically or using calculus to determine the number of solutions. This particular equation has a unique solution because the function \(f(x) = e^x - x - 2\) changes sign once, indicating a single crossing point on the graph.
### 4. Differential Equations
In the case of differential equations, uniqueness can be determined using the existence and uniqueness theorem (Picard-Lindelöf theorem). For example, a first-order ordinary differential equation \(dy/dx = f(x, y)\) with an initial condition \(y(x_0) = y_0\) has a unique solution if \(f(x, y)\) and \(\partial f / \partial y\) are continuous near \((x_0, y_0)\).
**Example:**
The differential equation \(dy/dx = 3y + 2x\) with \(y(0) = 1\) has a unique solution because the function and its partial derivative with respect to \(y\) are continuous.
### Conclusion
Whether a statement about a unique solution is true or false depends on the specific mathematical context. In linear systems, quadratic equations, nonlinear equations, and differential equations, different criteria apply to determine the uniqueness of solutions. The truth of the statement "a unique solution exists" is conditional and context-dependent.