https://file.aiquickdraw.com/mathbot/file/1728082190328m5hslfld.jpg

The system of equations provided in the image is:

\[
4x - 6y = 10y + 2 \tag{1}
\]
\[
ty = \frac{1}{2} + 2x \tag{2}
\]

We are asked to find the value of \( t \) such that the system has no solution. Let's proceed step by step:

### Step 1: Simplify the first equation

Starting with equation (1):

\[
4x - 6y = 10y + 2
\]

Rearrange to get all terms involving \( y \) on one side:

\[
4x = 16y + 2
\]

Now, solve for \( x \):

\[
x = 4y + \frac{1}{2}
\]

### Step 2: Substitute into the second equation

From equation (2):

\[
ty = \frac{1}{2} + 2x
\]

Substitute \( x = 4y + \frac{1}{2} \) into this equation:

\[
ty = \frac{1}{2} + 2 \left( 4y + \frac{1}{2} \right)
\]

Simplify:

\[
ty = \frac{1}{2} + 8y + 1
\]
\[
ty = 8y + \frac{3}{2}
\]

### Step 3: Analyze conditions for no solution

For the system to have no solution, the lines represented by these equations must be parallel, which means that the coefficients of \( y \) in both equations must be equal. From the first equation, we already know that:

\[
x = 4y + \frac{1}{2}
\]

So, the coefficient of \( y \) in equation (2) must also match the coefficient of \( y \) in the equation we derived. Therefore, we equate the coefficients of \( y \):

\[
t = 8
\]

Thus, the value of \( t \) that causes the system to have no solution is \( \boxed{8} \).

Would you like further details on any of these steps or have additional questions?

Here are five related questions you might explore:
1. How do we verify that two lines are parallel in general?
2. What happens when \( t \neq 8 \)? Does the system have one or infinitely many solutions?
3. How would the problem change if \( t \) was a function instead of a constant?
4. Can this method be extended to systems with three variables?
5. What conditions are needed for two lines to be perpendicular instead of parallel?

**Tip**: When dealing with systems of equations, always look at the relationship between the slopes of the lines to determine whether they are parallel, intersecting, or identical (coincident).

In the given system of equations, t is a constant. If the system has no solution, what is the value of t?

Learn how to find the value of t in a system of linear equations that results in no solution. This problem covers key algebraic concepts, including substitution method, linear systems, and the condition for parallel lines. Suitable for high school students in Grades 10-12.

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