Problem 30 - Algebra and Trigonometry

Question

\(29-34=\) Find parametric equations for the line with the given properties. Slope \(-2,\) passing through \((-10,-20)\)

Step-by-Step Solution

Answer

Parametric equations are \(x = -10 + t\) and \(y = -20 - 2t\).

Step 1: Understand the Components

A parametric equation for a line is typically expressed in the form:\[\begin{align*}x &= x_0 + at \y &= y_0 + bt\end{align*}\]where \((x_0, y_0)\) is a point on the line, \(a\) and \(b\) are directional numbers that relate to the slope, and \(t\) is the parameter. The given components here are the point \((-10, -20)\) and the slope \(m = -2\).

Step 2: Determine Directional Numbers

The slope \(-2\) implies a relationship between the change in \(y\) per change in \(x\). A common representation of the slope \(m = -2\) is \(\Delta y / \Delta x = -2/1\). Thus, for every increase of 1 in \(x\), \(y\) decreases by 2. Our directional numbers will be 1 in the \(x\)-direction and -2 in the \(y\)-direction, meaning \(a = 1\) and \(b = -2\).

Step 3: Formulate the Parametric Equations

Using the given point \((-10,-20)\) and the directional numbers \(a = 1\) and \(b = -2\), we can write the parametric equations:\[\begin{align*}x(t) &= -10 + 1\cdot t \y(t) &= -20 - 2\cdot t\end{align*}\]

Step 4: Simplify the Parametric Equations

Simplify these equations:\[\begin{align*}x(t) &= -10 + t \y(t) &= -20 - 2t\end{align*}\]These are the parametric equations for the line with the given slope and passing through the specified point.

Key Concepts

Slope

The concept of slope is pivotal in understanding how lines work in geometry. Slope is a measure of how steep a line is. Imagine a slide; the steeper it is, the higher the slope. Mathematically, it is defined as the ratio of the change in the vertical direction ( abla y) to the change in the horizontal direction ( abla x).
This ratio is commonly known as "rise over run."

• If the slope is positive, the line ascends as it moves to the right.
• If the slope is negative, like the -2 in our example, the line descends.

In this exercise, the given slope is -2, meaning for every unit that $x$ increases, $y$ decreases by 2 units. Thus, slope helps to determine the line's direction and steepness. Always remember, a larger absolute value of slope means a steeper line.

Directional Numbers

Directional numbers are crucial when dealing with parametric equations of a line. These numbers tell us how the line moves along the \(x\) and \(y\) axes as the parameter \(t\) changes. They represent the rate at which each coordinate of the line changes.
Given a slope, such as -2 in our case, directional numbers help us break it down practically:

• The slope is expressed as \(-2 = \frac{\Delta y}{\Delta x}\), which can be split into directional numbers: 1 for \(x\) and -2 for \(y\).
• This means for every 1 unit step of \(x\), the \(y\) changes by -2 units.

These directional numbers, referred to as \(a\) and \(b\), are then utilized to form the parametric equations that describe the line's path based on the parameter \(t\).

Point on a Line

To form parametric equations, a specific point on the line serves as an anchor. This point, given as $(-10, -20)$ in the exercise, is essential for determining the line's exact position.

• The point is denoted as $(x_0, y_0)$ in the parametric equation, acting as the starting point from where the line extends.
• We then use this point in combination with the directional numbers to express the line's equation.

In our situation, by substituting $(-10, -20)$ into the equations along with the directional numbers $a = 1$ and $b = -2$, we generate the parametric equations:
1. $x(t) = -10 + t$
2. $y(t) = -20 - 2t$
This point helps ensure the line reaches through the correct location on the $xy$ plane, and by varying $t$, the entire line can be traced.