To find the parametric equations for the line segment shown in the problem, let's start by analyzing the coordinates and the information provided.

Given Information:

1. The line segment begins at $(6,0)$ and ends at $(7,6)$.
2. The parameter $t$ starts at $t = 0$ and ends at $t = 6$.

Step-by-Step Solution:

We can use the following form for parametric equations for a line segment: $x = x_0 + (x_1 - x_0) \frac{t}{t_{end}}$ $y = y_0 + (y_1 - y_0) \frac{t}{t_{end}}$ where $(x_0, y_0)$ and $(x_1, y_1)$ are the starting and ending points, respectively, and $t_{end}$ is the maximum value of $t$.

Step 1: Define $x(t)$ and $y(t)$

Using the starting point $(6,0)$ and the ending point $(7,6)$, we substitute these values:

• $x_0 = 6$, $y_0 = 0$
• $x_1 = 7$, $y_1 = 6$
• $t_{end} = 6$

Step 2: Write the Parametric Equations

Now we can substitute into the formulas above:

1. For $x(t)$: $x(t) = 6 + (7 - 6) \frac{t}{6} = 6 + \frac{t}{6}$

2. For $y(t)$: $y(t) = 0 + (6 - 0) \frac{t}{6} = \frac{t}{1} = t$

Final Parametric Equations

The parametric equations that describe the line segment from $(6,0)$ to $(7,6)$ are: $x = 6 + \frac{t}{6}$ $y = t$ where $t$ varies from $0$ to $6$.

Would you like further explanation on any step, or do you have additional questions?

Relative Questions for Deeper Understanding

1. How would these parametric equations change if the starting and ending points were different?
2. What is the significance of the parameter $t$ in defining the path of the line segment?
3. How can you verify that the parametric equations are correct for a given segment?
4. How would the equations be modified if the line segment were defined in 3D?
5. What are other common forms of parametric equations for different types of curves?

Quick Tip

When writing parametric equations, make sure the range of $t$ aligns with the line segment's starting and ending points to capture the correct direction and length.