Question

Given three particular solutions \(\mathrm{y}, \mathrm{y}_{1}, \mathrm{y}_{2}\) of a linear equation. Prove that the expression \(\frac{y_{2}-y}{y-y_{1}}\) remains unchanged for any \(x\). What is the. geometrical significance of this result?

Short answer

Question: Prove that the expression \(\frac{y_{2}-y}{y-y_{1}}\) remains constant for all values of \(x\) for any linear equation, and interpret the geometrical significance of this result. Answer: The expression \(\frac{y_{2}-y}{y-y_{1}}\) remains constant for all values of \(x\) for any linear equation, as it simplifies to \(\frac{x_{2}-x}{x-x_{1}}\), which does not depend on \(y\). The geometrical significance of this result is that it represents the slope of the line passing through the points \((x_{1},y_{1})\), \((x,y)\), and \((x_{2},y_{2})\), indicating that the three points lie on the same straight line.

Step-by-step solution

Write down the general form of a linear equation

We know that the general form of a linear equation is given by: \(ax + by + c = 0\). We can rewrite this equation in terms of \(y\), which results in \(y = -\frac{a}{b}x - \frac{c}{b}\).

Substitute the particular solutions

We have three particular solutions of the linear equation: \(y, y_{1}, y_{2}\). Let's substitute these solutions into the equation derived in Step 1. We get: \(y = -\frac{a}{b}x - \frac{c}{b}, \quad y_{1} = -\frac{a}{b}x_{1} - \frac{c}{b}, \quad y_{2} = -\frac{a}{b}x_{2} - \frac{c}{b}\).

Rearrange the equations

Let's rearrange the three equations above to calculate the differences \(y_{2}-y\) and \(y-y_{1}\). For this, subtract the first equation from the third one and the first equation from the second one, respectively. \((y_{2}-y) = -\frac{a}{b}(x_{2}-x), \quad (y-y_{1}) = -\frac{a}{b}(x-x_{1})\).

Use the differences to form the desired expression

With the calculated differences in Step 3, we can now divide \((y_{2}-y)\) by \((y-y_{1})\): \(\frac{y_{2}-y}{y-y_{1}} = \frac{-\frac{a}{b}(x_{2}-x)}{-\frac{a}{b}(x-x_{1})}\).

Simplify the expression

Now we simplify the expression obtained in Step 4: \(\frac{y_{2}-y}{y-y_{1}} = \frac{x_{2}-x}{x-x_{1}}\). Since the expression does not contain any \(y\) terms, it will remain constant for any \(x\).

Determine the geometrical significance

The geometrical significance of this result can be understood by considering the coordinates \((x_{1},y_{1})\), \((x,y)\), and \((x_{2},y_{2})\) on a Cartesian plane. These three points lie on the same straight line since they are particular solutions of the linear equation. The expression \(\frac{y_{2}-y}{y-y_{1}}\) represents the slope of the line that passes through all three points and remains unaltered as \(x\) changes.

Key concepts

Particular Solutions of Linear Equations

Linear equations are fundamental in algebra and integral calculus, often representing relationships between variables in various scientific and mathematical contexts. A particular solution of a linear equation is a set of values for the variables that make the equation true. When working with linear equations, one typically encounters a standard form, such as ax + by + c = 0, where a, b, and c are constants, and x and y are variables.

When encountering three particular solutions, y, y_1, and y_2, we can substitute these into the linear equation's general form to obtain specific instances of the equation. The significance of showing that the expression (y_2-y)/(y-y_1) remains unchanged for any value of x demonstrates that, despite variations in the independent variable x, the ratio of the differences between particular solutions is constant. This demonstrates a unique property of linear equations—the slopes of the lines represented by each particular solution are equivalent.

Understanding particular solutions is crucial for solving more complex problems in calculus, where one might need to find specific values that satisfy both the linear differential equation and initial or boundary conditions. Simplified expressions resulting from applying particular solutions also allow for easier analysis and interpretation in applied mathematics.

Geometrical Significance in Calculus

The geometrical significance of algebraic expressions found in calculus can often illuminate a deeper understanding of the mathematical concepts involved. In the case of the linear expression (y_2-y)/(y-y_1), its geometrical significance lies in its representation of the slope of a line on the Cartesian coordinate system.

When we have a linear equation, and we plot the particular solutions on a graph, each solution will correspond to a point that lies on a straight line represented by the equation. The relation between these points is fundamental in understanding their geometric relation. Specifically, the expression we investigated shows that the slope calculated between any two points (x_1, y_1) and (x, y) is the same as that between points (x, y) and (x_2, y_2), indicating a single straight line through all three points.

In calculus, slopes of lines are important because they reflect rates of change and can indicate properties like the derivative of a function. Visualizing these algebraic concepts geometrically can aid students in building intuition and a more tangible understanding of the abstract mathematical principles they study.

Simplification of Algebraic Expressions

Simplification plays a crucial role in both algebra and calculus, as it makes equations more manageable and solutions clearer. Simplification of algebraic expressions involves reducing expressions to their simplest form through various algebraic operations like addition, subtraction, multiplication, division, and factoring.

In the given solution, simplification is used to transform the expression (y_2-y)/(y-y_1) by eliminating common factors and thus, revealing that the given algebraic fraction simplifies to a simple ratio of differences in the x variable. This process is achieved by identifying and canceling out common factors, which are the coefficients of the linear equation in this context. Simplifying algebraic expressions not only aids in finding solutions to equations but also makes it easier to understand the relationships between the variables involved.

For students, mastering simplification techniques is essential as it streamlines complex problems, making them more approachable and often elucidating underlying patterns or symmetries within the problem structure. Efficient simplification can also be important in higher-level calculus, especially when working with integrals or solving differential equations where a simplified expression can be integral to obtaining a solvable form.