Problem 19 - Contemporary Precalculus

Question

The given curve is part of the graph of an equation in \(x\) and \(y .\) Find the equation by eliminating the parameter. $$x=t-3, \quad y=2 t+1, \quad t \geq 0$$

Step-by-Step Solution

Answer

Question: Eliminate the parameter t from the parametric equations \(x = t - 3\) and \(y = 2t + 1\), and find the equation relating x and y. Answer: The equation relating x and y is \(y = 2x + 7\).

Step 1: Solve for t in one of the given equations

We are given the equation \(x = t - 3\). Now we want to solve this equation for t so that we can substitute it into the other given equation, which will help us eliminate the parameter t. Solving for t, we get: \(t = x + 3\)

Step 2: Plug the solved t value into the other given equation

We know that the other given equation is \(y = 2t + 1\). Now, as we have found the value of t in terms of x (\(t = x + 3\)), we can substitute this value into the given equation for y. Doing so, we get: \(y = 2(x + 3) + 1\)

Step 3: Simplify the equation to find the relationship between x and y

Now that we have an equation relating x and y, we can simplify it: \(y = 2(x + 3) + 1 \Rightarrow y = 2x + 6 + 1 \Rightarrow y = 2x + 7\) The equation relating x and y is \(y = 2x + 7\).

Key Concepts

Eliminate Parameter

Eliminating a parameter involves removing the parameter from parametric equations to obtain a direct relationship between variables, typically presented as one in terms of another. For the set of parametric equations given in our exercise, we have two equations:

• \( x = t - 3 \)
• \( y = 2t + 1 \)

Our goal is to eliminate the parameter \( t \). To do this, we start by solving one of the equations for \( t \). In the case of \( x = t - 3 \), we rearrange it to isolate \( t \):

• \( t = x + 3 \)

This expression for \( t \) can now be substituted into the second equation to remove the parameter \( t \) entirely. This is a crucial first step that allows us to directly relate \( x \) and \( y \). By substituting, we progress towards finding a relation that is entirely in terms of \( x \) and \( y \).

Relationship Between x and y

To find the relationship between \( x \) and \( y \) without the parameter \( t \), substitute the obtained expression for \( t \) from one equation into the other. We already derived that \( t = x + 3 \). Now, we substitute this \( t \) value into the \( y \) equation:

• Substitute \( t = x + 3 \) into \( y = 2t + 1 \):
• \( y = 2(x + 3) + 1 \)

This substitution step is essential as it transforms the parametric equations into a more conventional form. Now, \( y \) is described fully in terms of \( x \), which shows the direct relationship between them without referencing \( t \). Simplifying this expression further will yield a clearer understanding of how \( y \) depends on \( x \).

Equation in x and y

After substituting and simplifying, we arrive at the equation that represents the relationship between \( x \) and \( y \):

• Simplify \( y = 2(x + 3) + 1 \)
• \( y = 2x + 6 + 1 \)
• \( y = 2x + 7 \)

This simplified equation \( y = 2x + 7 \) is the result of eliminating the parameter \( t \). It captures the linear relationship between \( x \) and \( y \) derived from the original parametric equations.
In this form, the equation is familiar and useful for understanding the geometry of the graph it describes. For instance, it represents a line with a slope of 2 and a y-intercept of 7. Transitioning from parametric equations to this direct form makes it easier to visualize and apply within various mathematical contexts.