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. \((2,1)\) and \((3,5)\)

Step-by-Step Solution

Answer

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

Step 1: Determine the Direction Vector

To find the parametric equation of a line through two points, we first determine the direction vector. The direction vector \(\vec{d}\) is found by subtracting the coordinates of the first point from the second point: \((3-2, 5-1) = (1, 4)\). Thus, the direction vector is \((1, 4)\).

Step 2: Write the Parametric Equations of the Line

Using the point \((2,1)\) and the direction vector \((1,4)\), we write the parametric equations for the line as follows: \(x = 2 + t \\y = 1 + 4t \\), where \(t\) is the parameter.

Step 3: Solve for the Parameter t in Terms of x

To eliminate the parameter, we first solve one of the parametric equations for \(t\). Let's solve the equation \(x = 2 + t\) for \(t\): \(t = x - 2 \\).

Step 4: Substitute t into the y Equation

Now that we have \(t = x - 2\), substitute this expression for \(t\) into the parametric equation for \(y\): \(y = 1 + 4(x - 2) \\). This gives \(y = 1 + 4x - 8 \\), which simplifies to \(y = 4x - 7 \\).

Step 5: Rewrite in Standard Form

The standard form of a line is \(Ax + By = C\). Rearranging \(y = 4x - 7\) to match this form: \(-4x + y = -7 \\). Multiplying through by \(-1\) to make the coefficient of \(x\) positive, we get \(4x - y = 7\).

Key Concepts

Direction Vector

When working with parametric equations, one important concept is the direction vector. The direction vector gives us the direction in which the line extends. To find this vector, you take two points that the line passes through. Specifically, you subtract the coordinates of the first point from the coordinates of the second point. This process will give you the direction vector as a result.

For example, to find the direction vector of a line that passes through the points \((2, 1)\) and \((3, 5)\), you calculate:

• Subtract the x-coordinates: \(3 - 2 = 1\)
• Subtract the y-coordinates: \(5 - 1 = 4\)

Thus, the direction vector is \((1, 4)\). This vector helps to express the line in its parametric form, by showing how you must move from the initial point to stay on the line. Imagine it as a set of instructions: "Right 1, Up 4" from any point on the line to find another point.

Standard Form Equation

Trigonometry and algebra help us express geometric ideas in a structured form. In the context of lines, we often want the equation in what's called the "standard form." The standard form equation of a line is generally given as \(Ax + By = C\), where \(A\), \(B\), and \(C\) are integers, and \(A\) should be positive.

Starting with the parametric form of the line, like \(x = 2 + t\) and \(y = 1 + 4t\), you can transform it to the standard form. Initially write the explicit line equation, for instance, \(y = 4x - 7\). To transform this into standard form:

• Rearrange it to place all terms involving variables on one side: \-4x + y = -7\
• Multiply through by -1 to get the coefficient of \(x\) as positive: \4x - y = 7\

This process yields the standardized equation, making it easier to identify specific line properties and also useful when comparing with other lines.

Eliminating Parameter

Once we have a parametric equation of a line, one step often involves eliminating the parameter. This is generally done to convert parametric equations into the common types of line equations, like slope-intercept or standard form.

To eliminate the parameter, solve one of the parametric equations for the parameter \(t\). For example, if we have the equation \(x = 2 + t\), isolate \(t\): \[t = x - 2\]Substituting this expression into the other parametric equation, \(y = 1 + 4t\), gives a direct relation between \(x\) and \(y\). With the calculated value of \(t\), substitute:\[y = 1 + 4(x - 2)\]Which simplifies to:\[y = 4x - 7\]
Congratulations, you've eliminated the parameter! Now the equation is easier to interpret, as it directly shows you how \(y\) depends on \(x\). This form is especially useful for graphing and further analysis.