Problem 48 - Calculus for Biology and Medicine

Question

Find the parametric equation of the line in the \(x-y\) plane that goes through the given points. Then eliminate the parameter to find the equation of the line in standard form. $$ (2,1) \text { and }(3,5) $$

Step-by-Step Solution

Answer

The parametric equations are \(x = 2 + t\), \(y = 1 + 4t\). The line in standard form is \(4x - y = 7\).

Step 1: Find the Direction Vector

To find the parametric equation of a line, we first need a direction vector. Given two points, \((x_1, y_1)\) and \((x_2, y_2)\), the direction vector \(\mathbf{d}\) can be found as \((x_2 - x_1, y_2 - y_1)\). Substitute the points \((2, 1)\) and \((3, 5)\) into the equation: \(\mathbf{d} = (3 - 2, 5 - 1) = (1, 4)\).

Step 2: Write the Parametric Equations

With a starting point \((2,1)\) and a direction vector \((1,4)\), the parametric equations for the line can be formulated as \(x = 2 + t\) and \(y = 1 + 4t\), where \(t\) is the parameter.

Step 3: Eliminate the Parameter

To find the equation of the line in standard form, eliminate the parameter \(t\) from the parametric equations. Solve for \(t\) in terms of \(x\): \(x = 2 + t \Rightarrow t = x - 2\). Substitute \(t\) in the equation for \(y\): \(y = 1 + 4(x - 2)\).

Step 4: Simplify the Expression

Distribute the 4 and then simplify: \(y = 1 + 4x - 8\), which simplifies to \(y = 4x - 7\).

Step 5: Write the Equation in Standard Form

The standard form of a line is \(Ax + By = C\). Rearrange the simplified equation to \(4x - y = 7\), which is the standard form of the line.

Key Concepts

Direction Vector

To understand how to find the parametric equation of a line, it's important to start with the direction vector. A direction vector aims to point out the direction in which the line extends. This vector is derived using two given points that lie on the line, in this case, (2,1) and (3,5).

The direction vector \(\mathbf{d} = (x_2 - x_1, y_2 - y_1)\) is calculated by subtracting the corresponding coordinates of these two points. For the given points, the calculation becomes:

• The direction vector is \(\mathbf{d} = (3 - 2, 5 - 1) = (1, 4)\)

This vector \( (1, 4) \) gives both the x and y components of the direction in which the line spreads. By knowing this vector, you have the essential part needed to write the parametric equations.

Standard Form

When we need the standard form of a line's equation, we aim for a neat, well-recognized format. The standard form appears as \(Ax + By = C\). This form has specific coefficients and constants where both \(A\) and \(B\) should ideally be integers.

In this exercise, after framing the line with parametric equations using a start point and a direction vector, we move to convert this into standard form.

• Start off with the equation in parametric terms, such as \(x = 2 + t\) and \(y = 1 + 4t\).
• Rearrange the parametric equation to eliminate the parameter \(t\) (which we will discuss in the next section) and simplify it to get \(y = 4x - 7\).
• To change this into the standard form, rearrange it to \(4x - y = 7\).

In these steps, the transformation from parametric equations to standard form is complete, following the basic properties of linear algebra.

Eliminate Parameter

Eliminating the parameter \(t\) is a key step in transitioning from a parametric equation to a coordinate-based, standard form. The process involves expressing \(t\) in terms of one coordinate, and then substituting it into the other equation. Let's break it down for clarity.

• Start with the parametric equations: \(x = 2 + t\) and \(y = 1 + 4t\).
• Isolate \(t\) in one equation. From \(x = 2 + t\), we get \(t = x - 2\).
• Substitute \(t\) from this expression into the equation for \(y\). This becomes \(y = 1 + 4(x - 2)\).
• Finally, simplify the expression: expand and combine terms to get \(y = 4x - 7\).

By eliminating the parameter, you transform the line description into a format that is universally recognizable, where \(x\) and \(y\) relate directly, rather than each to an external parameter. This step solidifies understanding and ensures the line is presented in a form usable in various further applications.