Problem 44 - Algebra and Trigonometry

Question

The following are parametric equations of the line through \(\left(x_{1}, y_{1}\right)\) and \(\left(x_{2}, y_{2}\right)\) $$ x=x_{1}+t\left(x_{2}-x_{1}\right) \quad \text { and } \quad y=y_{1}+t\left(y_{2}-y_{1}\right) $$ Eliminate the parameter and write the resulting equation in point-slope form.

Step-by-Step Solution

Answer

The point-slope form of the line is \( y-y_{1} = \frac{y_{2}-y_{1}}{x_{2}-x_{1}} (x - x_{1}) \)

Step 1: Express the parameter \( t \) from each equation

Isolate \( t \) in each of the given equations as follows:\n- For \( x=x_{1}+t\left(x_{2}-x_{1}\right) \), we get \( t=\frac{x-x_{1}}{x_{2}-x_{1}} \)- For \( y=y_{1}+t\left(y_{2}-y_{1}\right) \), we get \( t=\frac{y-y_{1}}{y_{2}-y_{1}} \)

Step 2: Setting the expressions for \( t \) equal

Since \( t \) is the same in both equations, we can set the two expressions for \( t \) equal to each other, as follows:\n\( \frac{x-x_{1}}{x_{2}-x_{1}} = \frac{y-y_{1}}{y_{2}-y_{1}} \)

Step 3: Rearrange to point-slope form

We can rearrange this equation to the point-slope form of a line (y - y1 = m(x - x1)), where \( m \) is the slope of the line:\n\( y-y_{1} = \frac{y_{2}-y_{1}}{x_{2}-x_{1}} (x - x_{1}) \)

Key Concepts

Point-Slope Form

The point-slope form is an equation used to describe a straight line on a graph. It is particularly useful when you know one point on the line and the slope of the line. The general formula is:

• \( y - y_1 = m(x - x_1) \)

Here,

• \( (x_1, y_1) \) is a point on the line,
• \( m \) is the slope of the line.

To use this form, replace \( x_1 \) and \( y_1 \) with the coordinates of your given point. Replace \( m \) with the slope you have calculated. This form of a linear equation emphasizes the line's slope and provides a simple way to build the line equation from a given point. It makes it straightforward to identify both the slope and one point through which the line passes.

Eliminate the Parameter

When working with parametric equations, "eliminating the parameter" means removing the parameter to rewrite the equation solely in terms of \( x \) and \( y \). It is a crucial step to derive the cartesian equation of a curve given in parametric form. For example, given parametric equations:

• \( x = x_1 + t(x_2 - x_1) \)
• \( y = y_1 + t(y_2 - y_1) \)

Here, \( t \) is the parameter. To eliminate it:

• Isolate \( t \) in both equations.
• Set the two expressions for \( t \) equal to one another.

Thus, you obtain an equation solely in \( x \) and \( y \). This step allows you to express the relationship between \( x \) and \( y \) (the line) directly, without the extra parameter. It's essential for converting parametric equations to their more familiar algebraic form.

Slope of the Line

The slope of a line is a measure of its steepness and direction. It describes how much \( y \) increases or decreases for a unit increase in \( x \). Mathematically, the slope \( m \) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is calculated as:

• \( m = \frac{y_2 - y_1}{x_2 - x_1} \)

This formula provides the rise-over-run ratio, showing the vertical change (delta \( y \)) over the horizontal change (delta \( x \)). In the context of parametric equations, once you set the expressions for \( t \) equal and rearrange, you will note this formula at play as it shapes the point-slope equation.

• Positive slope: Line rises as it moves from left to right.
• Negative slope: Line falls as it moves from left to right.
• Zero slope: Horizontal line.
• Undefined slope: Vertical line.

The slope is a pivotal concept in understanding linear relationships and forms the backbone of equations defining these lines.