Problem 39 - Essential Calculus

Question

Find two different sets of parametric equations for the rectangular equation. $$ y=3 x-2 $$

Step-by-Step Solution

Answer

The first set of parametric equations for the rectangular equation \(y=3x-2\) are \(x = t\) and \(y = 3t - 2\). The second set of parametric equations are \(x =(t+2)/3\) and \(y = t\).

Step 1: First set of Parametric Equations

We can choose \(t\) to be \(x\) in the given equation and simply calculate \(y\). We let \(x = t\) and \(y = 3t - 2\). Therefore, our first set of parametric equations will be \(x = t\) and \(y = 3t - 2\).

Step 2: Second set of Parametric Equations

For the second set, let's choose to vary \(y\) instead of \(x\). We can let \(t\) be \(y+2\), therefore \(y=t\) and \(x=(t+2)/3\). Therefore, our second set of parametric equations will be \(x =(t+2)/3\) and \(y = t\).

Key Concepts

Rectangular Equation

In coordinate geometry, a rectangular equation represents a relationship between the variables \(x\) and \(y\) on a Cartesian coordinate plane. They are often the kind of equations we encounter in algebra classes. Rectangular equations are useful for defining curves and lines using straightforward algebraic expressions. For instance, the linear equation \(y = 3x - 2\) describes a straight line with a slope of 3 and a y-intercept of -2. This is a simple form where we directly relate \(x\) and \(y\) using arithmetic operations.

• The equation \(y = 3x - 2\) depicts a line with a slope (rise over run) of 3.
• The line crosses the y-axis at the point (0, -2), which is the y-intercept.

Rectangular equations provide direct insight into the geometry of a line or curve, making them easy to work with for plotting graphs or conducting algebraic manipulations. Their form is simple, making them an excellent starting point for more complex functions.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, involves using coordinate systems to geometrically represent algebraic equations. This process allows us to visualize equations by plotting them on graph paper, typically using the Cartesian coordinate system. The Cartesian system uses two perpendicular axes: the x-axis (horizontal) and the y-axis (vertical), forming a plane to plot points.

• Every point on this plane is represented by pairs of numbers \((x, y)\).
• It allows the interpretation of geometric properties using algebra.

For the equation \(y = 3x - 2\), plotting it in the coordinate plane demonstrates its geometric meaning as a straight line. The slope and y-intercept become visible, helping learners understand the relationship between the algebraic expression and its geometric representation. Understanding coordinate geometry is fundamental for translating between different types of equations, such as converting between rectangular and parametric equations.

Equation Conversion

Equation conversion, particularly transitioning from rectangular to parametric form, involves expressing a relationship in terms of a third variable, often called a parameter, typically denoted as \(t\). This practice is convenient in various mathematical applications such as kinematics, where it's necessary to describe the motion of objects separately over time.

• In the given equation \(y = 3x - 2\), one can introduce a parameter \(t\) to solve the equation differently.
• For example, set \(x = t\) and solve for \(y\) to get \(y = 3t - 2\).
• Alternatively, by letting \(y = t\) and solving for \(x\), it transforms to \(x = (t + 2)/3\).

Through parametric equations, one can describe the same line with different sets of equations, each defined through a parameter \(t\). This switching of variables provides flexibility and a deeper understanding of how points along the line relate to one another in terms of their movement through the variable \(t\). These transformations reveal multiple perspectives on the same geometric entity, enhancing our mathematical toolbox.