Question

Give a set of parametric equations that generates the line with slope -2 passing through (1,3)

Short answer

Answer: The parametric equations for the given line are x(t) = t and y(t) = -2t + 5.

Step-by-step solution

Find the line equation

Given the slope (m) and a point (x1, y1) on the line, we can find the y-intercept (b) using the following formula: b = y1 - m * x1 In this case, we are given m = -2 and the point (1, 3): b = 3 - (-2)*1 b = 3 + 2 b = 5 Now, we can write down our line equation: y = -2x + 5

Convert the line equation to parametric equations

To obtain the parametric equations from the line equation, let t be a parameter and make the x-coordinate the t value. We then get x = t and y = -2t + 5. Given the line equation, our parametric equations are as follows: x(t) = t y(t) = -2t + 5 Thus, our final set of parametric equations is: x(t) = t y(t) = -2t + 5

Key concepts

Line Equation

A line equation in the Cartesian coordinate system represents a straight line. The standard form of a line equation is expressed as \( y = mx + b \). Here, \( m \) is the slope of the line, which indicates its steepness and direction, while \( b \) is the y-intercept, which shows where the line crosses the y-axis. If you know a point \( (x_1, y_1) \) on the line and the slope \( m \), you can find \( b \) using the formula \( b = y_1 - mx_1 \).
Imagine you have a line passing through the point (1, 3) with a slope of -2 as in the original exercise. Start by substituting these values into the formula to calculate the intercept:\

• Point: \( (1, 3) \)
• Slope (\( m \)): -2

Substitute to get \( b \):

• \( b = 3 - (-2) \cdot 1 = 3 + 2 = 5 \)

Therefore, the line equation becomes \( y = -2x + 5 \). This equation describes the relationship between \( x \) and \( y \) along the line. It shows that for every unit increase in \( x \), \( y \) decreases by 2 units because the slope is -2. The line crosses the y-axis at (0,5).

Slope

The slope of a line is a measure of its steepness or incline. It's a crucial part of the line equation \( y = mx + b \), where \( m \) represents the slope. The slope tells us how much \( y \) changes for every change in \( x \). A positive slope means the line rises as \( x \) increases, whereas a negative slope means the line falls.
Understanding the slope helps in visualizing the inclination of the line upon a graph. If you see a slope of -2, as given in the original exercise, this means:

• The line falls leftward at a rate of 2 units for each 1 unit increase in \( x \).
• The line is slanting downwards, indicating a negative correlation between \( x \) and \( y \).

To represent a slope in parametric form, we can derive it from the line equation. By using a parameter, \( t \), we express both \( x \) and \( y \) in terms of \( t \). In the original solution, with the slope -2, the parametric equations become \( x(t) = t \) and \( y(t) = -2t + 5 \). This representation is dynamic, meaning by adjusting \( t \), you trace the line's path.

Coordinate System

A coordinate system is a method for identifying the position of points in space. The most common is the Cartesian coordinate system, consisting of perpendicular axes: the x-axis (horizontal) and the y-axis (vertical). Each point is defined by an ordered pair \( (x, y) \), where \( x \) is the horizontal distance from the origin and \( y \) is the vertical distance.
In the context of the original exercise, the Cartesian coordinate system allows us to map the line corresponding to the equation \( y = -2x + 5 \). Understanding this system is crucial when converting line equations to parametric equations. Parametric equations like \( x(t) = t \) and \( y(t) = -2t + 5 \) describe a line differently by parameterizing it, typically with \( t \) representing time or another varying quantity.
The parametric form is useful for finding specific points on the line by substituting different values of \( t \). Each replacement gives a new point \( (x(t), y(t)) \) on the line in the coordinate system. For example, if \( t = 1 \), then \( x(1) = 1 \) and \( y(1) = 3 \), which verifies that the given point (1, 3) lies on the line. This visualization through a coordinate system allows for deeper understanding and application of linear equations.