Question

All lines are in the \((x, y)\) plane. Find the slope of the line whose paramerric equation is \(r=(i-j)+(2 i+3 j) t\).

Short answer

The slope is \( \frac{3}{2} \).

Step-by-step solution

Convert Parametric Form to Cartesian Form

The given parametric equations are: \[ x = 1 + 2t \]\[ y = -1 + 3t \]. To find the slope, convert these into the form of a Cartesian equation (y = mx + c).

Express t in Terms of x

From the equation \[ x = 1 + 2t \], solve for t:\[ t = \frac{x - 1}{2} \].

Substitute t into y Equation

Substitute \[ t = \frac{x - 1}{2} \] into \[ y = -1 + 3t \]:\[ y = -1 + 3\left(\frac{x - 1}{2}\right) \]\[ y = -1 + \frac{3(x - 1)}{2} \]\[ y = -1 + \frac{3x - 3}{2} \]\[ y = -1 + \frac{3x}{2} - \frac{3}{2} \]\[ y = \frac{3x}{2} - \frac{5}{2} \].

Identify the Slope

In the equation \[ y = \frac{3x}{2} - \frac{5}{2} \], identify the coefficient of x as the slope. The slope is \( \frac{3}{2} \).

Key concepts

Parametric Equations

In mathematics, parametric equations involve using a third variable, often denoted as \( t \) (parameter), to define both \( x \) and \( y \) coordinates of a point. This is useful for describing curves in the \(( x, y )\) plane.

For example, the given parametric equations are:

• \( x = 1 + 2t \)
• \( y = -1 + 3t \)

These equations articulate how both \( x \) and \( y \) change with respect to \( t \). With parametric equations, we can describe more complex curves compared to traditional Cartesian equations. This method is highly beneficial in fields like physics and engineering where motion and various trajectories need to be accounted for.

Slope Calculation

The slope of a line in the Cartesian plane expresses how steep the line is. Mathematically, the slope \(m\) is defined as \( m = \frac{dy}{dx} \). For parametric equations, we need to use the derivatives of \( x(t) \) and \( y(t) \) with respect to \( t \).

Given the parametric equations:

• \( x = 1 + 2t \)
• \( y = -1 + 3t \)

We can find the derivatives:

• \( \frac{dx}{dt} = 2 \)
• \( \frac{dy}{dt} = 3 \)

Hence, the slope \( m \) is computed as:
\( m = \frac{dy/dt}{dx/dt} = \frac{3}{2} \)
This method provides a straightforward way to determine the slope by using derivative ratios. However, in our exercise, we converted parametric equations into Cartesian form and then identified the slope from the resulting linear equation.

Cartesian Form Conversion

To convert parametric equations to the Cartesian form (\( y = mx + c \)), we follow systematic steps to eliminate the parameter \( t \). This gives us an equation purely in terms of \( x \) and \( y \).

Begin with the parametric forms:

• \( x = 1 + 2t \)
• \( y = -1 + 3t \)

First, solve for \( t \) in terms of \( x \):
\( t = \frac{x - 1}{2} \)

Next, substitute this expression into the equation for \( y \):
\( y = -1 + 3\left(\frac{x - 1}{2}\right) \)
\( y = -1 + \frac{3(x - 1)}{2} \)
\( y = -1 + \frac{3x - 3}{2} \)
\( y = \frac{3x}{2} - \frac{5}{2} \)

This is now in Cartesian form \( y = mx + c \), where \( m = \frac{3}{2} \) is the slope and \( c = -\frac{5}{2} \) is the y-intercept. The process of conversion makes it easier to apply standard methods for calculating geometric properties like slopes and intersections.