91影视

A system of equations is a collection of two or more equations with the same set of variables. These systems are an integral part of linear algebra. In the given problem, we are working with a simple system consisting of two linear equations:

• \( x_1 + 3x_2 = 2 \)
• \( 3x_1 + hx_2 = k \)

These equations represent lines in a two-dimensional plane. The task is to find scenarios for the system that will result in different types of solutions: no solution, a unique solution, or many solutions. Understanding these possibilities is central to analyzing and solving systems of linear equations effectively.

A unique solution occurs when exactly one set of values satisfies all equations in the system simultaneously. For our system of equations, this means the two lines represented by the equations intersect at exactly one point. Mathematically, this is represented by a non-zero determinant of the matrix formed by the coefficients of the variables.For the given matrix:\[\begin{bmatrix} 1 & 3 \ 3 & h \end{bmatrix}\]The determinant is calculated as \( h - 9 \). To ensure a unique solution exists, the determinant must not equal zero, which implies \( h eq 9 \). In this case, there is no restriction on the value of \( k \), as it does not affect the condition for a unique solution. By setting \( h = 5 \), for example, a unique solution is guaranteed, as the determinant becomes \( 5 - 9 = -4 \), which is non-zero.

A system of equations has no solution when the equations represent parallel lines that never intersect. For two lines to be parallel, they must have the same slope but different intercepts. In our system, the slope of the first equation is \( -\frac{1}{3} \).Similarly, for the second equation, \( \begin{aligned} 3x_1 + hx_2 = k \end{aligned} \), the slope must also be \( -\frac{1}{3} \) to result in parallel lines. Solving \( \frac{3}{h} = \frac{1}{3} \), we find \( h = 9 \). However, to ensure no intersection, the intercept must differ from that of the first line. Thus, the second equation's intercept \( k \) must not equal \( 6 \), making the condition \( k eq 6 \). This ensures that the lines remain parallel and do not coincide, leading to no solution.

The system of equations has many solutions when the equations represent the same line, meaning they overlap completely. For this to happen, both the slope and the intercept of the lines must be identical.For our equations, this means setting \( h = 9 \) for the slopes to be equal (\( -\frac{1}{3} \)). In addition, the intercepts must also match by setting \( k = 6 \). When both these conditions are satisfied, the lines are coincident, or exactly the same, leading to infinitely many solutions. This means any point on the line is a solution to both equations, highlighting the concept that some systems can have numerous solutions if the constraints align perfectly.