15) Graph the line using the parametric equations
x = 1 + 2t, y = 3 + t

To graph the line using the parametric equations \( x = 1 + 2t \) and \( y = 3 + t \), let's break it down step by step.

### Step 1: Understand the parametric equations
- The equation for \( x \) is \( x = 1 + 2t \).
- The equation for \( y \) is \( y = 3 + t \).

Here, \( t \) is the parameter, and as \( t \) varies, the coordinates \( (x, y) \) trace out a line on the Cartesian plane.

### Step 2: Eliminate the parameter \( t \) (optional)
We can eliminate \( t \) to find the Cartesian equation of the line.

1. Solve the equation for \( y \) to express \( t \) in terms of \( y \):
   \[
   y = 3 + t \implies t = y - 3
   \]

2. Substitute this expression for \( t \) into the equation for \( x \):
   \[
   x = 1 + 2t = 1 + 2(y - 3)
   \]
   Simplify this equation:
   \[
   x = 1 + 2y - 6 \implies x = 2y - 5
   \]
   The Cartesian form of the equation is:
   \[
   y = \frac{1}{2}x + \frac{5}{2}
   \]

### Step 3: Plot points on the graph
Let's choose a few values of \( t \) to find corresponding points \( (x, y) \):

- For \( t = 0 \): 
  \[
  x = 1 + 2(0) = 1, \quad y = 3 + 0 = 3
  \]
  Point: \( (1, 3) \)

- For \( t = 1 \): 
  \[
  x = 1 + 2(1) = 3, \quad y = 3 + 1 = 4
  \]
  Point: \( (3, 4) \)

- For \( t = -1 \): 
  \[
  x = 1 + 2(-1) = -1, \quad y = 3 - 1 = 2
  \]
  Point: \( (-1, 2) \)

### Step 4: Draw the line
Plot the points \( (1, 3) \), \( (3, 4) \), and \( (-1, 2) \) on the Cartesian plane. Then, draw a straight line through these points.

### Summary of the Line
- The parametric equations define a line.
- The Cartesian equation of the line is \( y = \frac{1}{2}x + \frac{5}{2} \).
- Points like \( (1, 3) \), \( (3, 4) \), and \( (-1, 2) \) lie on this line.

Would you like a detailed graph of this line? 

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### Follow-up Questions:
1. How do you find the slope of the line from the parametric equations?
2. What does the parameter \( t \) represent geometrically?
3. How do you find the direction vector of the line using parametric equations?
4. Can you determine the intercepts of the line from the Cartesian equation?
5. How would you describe the motion along the line as \( t \) increases?

#### Tip:
When plotting parametric equations, choosing a range of \( t \) values helps in understanding the direction and extent of the line or curve being plotted.

Learn how to graph a line using parametric equations x = 1 + 2t and y = 3 + t. This problem involves understanding parametric equations, converting them to Cartesian form, and plotting points on the Cartesian plane. Suitable for high school students studying parametric equations and linear graphing.

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