indeterminate coefficient

简明释义

未定系数

英英释义

例句

1.In solving the equation, we found that the term with the indeterminate coefficient (不确定系数) could take on multiple values depending on the other variables.

在解这个方程时，我们发现包含不确定系数的项(indeterminate coefficient)可以根据其他变量取多个值。

2.In our research, we identified an indeterminate coefficient (不确定系数) that influenced the outcome of the experiment.

在我们的研究中，我们识别出一个影响实验结果的不确定系数(indeterminate coefficient)。

3.When modeling the system, we had to account for the indeterminate coefficient (不确定系数) in our equations.

在建模系统时，我们必须在方程中考虑到不确定系数(indeterminate coefficient)。

4.The indeterminate coefficient (不确定系数) in the linear regression model made it difficult to predict future trends accurately.

线性回归模型中的不确定系数(indeterminate coefficient)使得准确预测未来趋势变得困难。

5.The presence of an indeterminate coefficient (不确定系数) complicates the analysis of the polynomial function.

存在不确定系数(indeterminate coefficient)使得多项式函数的分析变得复杂。

作文

In mathematics, particularly in the field of algebra, the concept of an indeterminate coefficient is crucial for understanding how to solve equations that involve unknown variables. An indeterminate coefficient refers to a coefficient that cannot be determined without additional information or constraints. This idea often arises in polynomial equations or when dealing with systems of linear equations. To illustrate this, consider a simple linear equation such as ax + b = 0, where 'a' and 'b' are coefficients. If 'a' is zero, the equation becomes b = 0, which can either have no solution or infinitely many solutions depending on the value of 'b'. This situation highlights the role of indeterminate coefficients because the value of 'a' affects the outcome of the equation. In more complex scenarios, such as solving a system of equations, the presence of indeterminate coefficients can lead to multiple solutions. For instance, if we have two equations: 2x + 3y = 6 and 4x + 6y = 12, the second equation is merely a multiple of the first, indicating that there are infinitely many solutions along the line defined by these equations. Here, the coefficients of 'x' and 'y' can be considered indeterminate coefficients since they do not provide unique solutions but rather a family of solutions. Understanding indeterminate coefficients is not only essential for solving mathematical problems but also has practical applications in various fields such as physics, engineering, and economics. For example, in physics, when modeling motion, certain parameters like friction or air resistance may be treated as indeterminate coefficients until specific conditions are known. Moreover, in statistics, when performing regression analysis, the coefficients of the independent variables can sometimes be indeterminate coefficients if there is multicollinearity among the variables. This means that the variables are highly correlated, making it difficult to ascertain the individual effect of each variable on the dependent variable. In such cases, analysts must carefully interpret the results, recognizing that the coefficients are not fixed but rather dependent on the data used. In conclusion, the concept of indeterminate coefficients plays a pivotal role in both theoretical and applied mathematics. It encourages critical thinking and problem-solving skills as one navigates through equations and systems that do not yield straightforward answers. By acknowledging the nature of indeterminate coefficients, mathematicians and scientists can better approach complex problems, ensuring that they consider all possible solutions and the implications of varying coefficients. This understanding ultimately leads to more robust conclusions and advancements in various disciplines, showcasing the importance of mastering concepts like indeterminate coefficients in mathematical education and beyond.

在数学中，尤其是在代数领域，不确定系数的概念对于理解如何解决涉及未知变量的方程至关重要。不确定系数是指一个系数，在没有额外信息或约束的情况下无法确定。这个概念通常出现在多项式方程中或处理线性方程组时。为了说明这一点，考虑一个简单的线性方程，比如 ax + b = 0，其中 'a' 和 'b' 是系数。如果 'a' 为零，则方程变为 b = 0，这可能没有解或根据 'b' 的值有无穷多个解。这种情况突显了 不确定系数 的作用，因为 'a' 的值会影响方程的结果。在更复杂的情况下，比如解决方程组时，不确定系数 的存在可能导致多个解。例如，如果我们有两个方程：2x + 3y = 6 和 4x + 6y = 12，第二个方程只是第一个方程的倍数，表明沿着这些方程定义的直线有无穷多个解。在这里，'x' 和 'y' 的系数可以视为 不确定系数，因为它们并未提供唯一的解，而是提供了一系列解。理解 不确定系数 不仅对解决数学问题至关重要，还在物理、工程和经济等多个领域具有实际应用。例如，在物理学中，当建模运动时，某些参数如摩擦或空气阻力可能被视为 不确定系数，直到已知特定条件。此外，在统计学中，当进行回归分析时，自变量的系数有时可能是 不确定系数，如果变量之间存在多重共线性。这意味着这些变量高度相关，使得难以确定每个变量对因变量的单独影响。在这种情况下，分析师必须仔细解释结果，认识到这些系数不是固定的，而是依赖于所使用的数据。总之，不确定系数 的概念在理论和应用数学中都发挥着关键作用。它鼓励批判性思维和解决问题的能力，因为人们在处理不提供直接答案的方程和系统时，能够更好地应对。通过承认 不确定系数 的性质，数学家和科学家能够更好地应对复杂问题，确保他们考虑所有可能的解决方案及不同系数的影响。这种理解最终导致更为稳健的结论和各学科的进步，展示了掌握像 不确定系数 这样的概念在数学教育及其他领域的重要性。