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To solve for the coordinates of the intersection point \( T \) of lines AC and BD, we'll follow these steps:

### Given equations:

1. \( 3x + 5y = 16 \) (Equation of line AC)
2. \( x - 6y = -10 \) (Equation of line BD)

### Step 1: Solve the system of equations.

We will use the substitution method or the elimination method to solve for \( x \) and \( y \). Let's proceed with the **substitution method** by solving the second equation for \( x \).

From the second equation:

\[
x = 6y - 10
\]

### Step 2: Substitute \( x = 6y - 10 \) into the first equation.

Now, substitute this expression for \( x \) into the first equation \( 3x + 5y = 16 \):

\[
3(6y - 10) + 5y = 16
\]

Simplifying:

\[
18y - 30 + 5y = 16
\]

\[
23y - 30 = 16
\]

\[
23y = 46
\]

\[
y = 2
\]

### Step 3: Find \( x \).

Now substitute \( y = 2 \) back into the expression for \( x \):

\[
x = 6(2) - 10 = 12 - 10 = 2
\]

### Conclusion:

The intersection point \( T \) has coordinates \( (x, y) = (2, 2) \).

Let me know if you'd like further details or if you have any questions! Here are some follow-up questions to expand the problem:

1. What method can we use besides substitution to solve the system of equations?
2. How can we verify that \( (2, 2) \) satisfies both equations?
3. What is the significance of the intersection point in terms of geometry?
4. How would the solution change if the equations had no intersection point (parallel lines)?
5. Can we use matrices to solve this system of equations?

**Tip:** Always double-check your solution by substituting back into the original equations.

Find the coordinates of the intersection point T of lines AC and BD.

Learn how to solve a system of linear equations using the substitution method to find the intersection point of two lines. This problem involves solving the equations 3x + 5y = 16 and x - 6y = -10, suitable for students in Grades 9-12.

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