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We are tasked with solving two parts for this problem involving two lines \( p \) and \( q \).

### Given:
- Line \( p \) passes through the points \( (1, -3) \) and \( (4, 6) \).
- Line \( q \) has the equation \( y = \frac{4}{5}x + 5 \).
- The lines intersect at point \( T \).

### Part (a): Find the equation for line \( p \)
To find the equation of line \( p \), we will:
1. Determine the slope of the line using the two given points.
2. Use the slope-point form to find the equation.

#### Step 1: Slope of line \( p \)
The slope formula is:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]
Using the points \( (1, -3) \) and \( (4, 6) \):
\[
m = \frac{6 - (-3)}{4 - 1} = \frac{6 + 3}{3} = \frac{9}{3} = 3
\]
Thus, the slope of line \( p \) is 3.

#### Step 2: Equation of line \( p \)
Using the point-slope form:
\[
y - y_1 = m(x - x_1)
\]
Substituting \( m = 3 \) and the point \( (1, -3) \):
\[
y + 3 = 3(x - 1)
\]
Simplifying:
\[
y + 3 = 3x - 3
\]
\[
y = 3x - 6
\]
Thus, the equation of line \( p \) is \( y = 3x - 6 \).

---

### Part (b): Find the coordinates of point \( T \)
To find the coordinates of point \( T \), we need to solve for the intersection of lines \( p \) and \( q \).

#### Line equations:
- Line \( p \): \( y = 3x - 6 \)
- Line \( q \): \( y = \frac{4}{5}x + 5 \)

We set the two equations equal to each other to find \( x \):
\[
3x - 6 = \frac{4}{5}x + 5
\]
Multiplying through by 5 to eliminate the fraction:
\[
5(3x - 6) = 5\left(\frac{4}{5}x + 5\right)
\]
\[
15x - 30 = 4x + 25
\]
Now, collect like terms:
\[
15x - 4x = 25 + 30
\]
\[
11x = 55
\]
\[
x = 5
\]

Now substitute \( x = 5 \) back into one of the original equations to find \( y \). Using \( y = 3x - 6 \):
\[
y = 3(5) - 6 = 15 - 6 = 9
\]

Thus, the coordinates of point \( T \) are \( (5, 9) \).

---

### Final answers:
1. The equation of line \( p \) is \( y = 3x - 6 \).
2. The coordinates of the intersection point \( T \) are \( (5, 9) \).

Do you have any questions or would you like further details on any steps?

### Relative Questions:
1. How do you find the equation of a line given two points?
2. What is the geometric meaning of the slope of a line?
3. How do you solve a system of linear equations?
4. What does it mean when two lines intersect?
5. Can two non-parallel lines in a plane never intersect?

### Tip:
When solving for the intersection of two lines, always check if they are parallel first (i.e., have the same slope) to avoid unnecessary calculations.

The line p passes through (1, -3) and (4, 6) and the line q has equation y = (4/5)x + 5. The lines p and q intersect at T. (a) Find an equation for p. (b) Find the coordinates of T.

Learn how to find the equation of a line passing through two points and determine the intersection of two lines with given equations. This problem focuses on key concepts in coordinate geometry, including finding slopes and solving systems of linear equations.

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