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.