Okay, let's see. I need to find the equation of a line in the form y = mx + b, given the parameterized form. The parameterization is given as:

\[
\begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} 8 \\ -1 \end{pmatrix} + t \begin{pmatrix} 2 \\ 3 \end{pmatrix}
\]

So, the line is expressed in terms of a parameter t. Each component of the vector (x, y) is given as a function of t. The first component is 8 + 2t, and the second component is -1 + 3t. Therefore, x = 8 + 2t and y = -1 + 3t.

To find the equation of the line in slope-intercept form (y = mx + b), I need to eliminate the parameter t and express y in terms of x. Let's start by solving the equation for x to find t in terms of x, and then substitute that into the equation for y.

Starting with the x component:

x = 8 + 2t

I can subtract 8 from both sides to get:

x - 8 = 2t

Then divide both sides by 2:

t = (x - 8)/2

Now that I have t expressed in terms of x, I can substitute this into the equation for y:

y = -1 + 3t

Substituting t:

y = -1 + 3 * ((x - 8)/2)

Simplify that:

First, multiply 3 by (x - 8)/2. Let's compute that:

3*(x - 8)/2 = (3x - 24)/2

So, the equation becomes:

y = -1 + (3x - 24)/2

Now, let's combine the constants. To do that, I need to express -1 as a fraction with denominator 2:

-1 = -2/2

So, substituting that in:

y = (-2/2) + (3x - 24)/2

Combine the terms:

y = (3x - 24 - 2)/2

Simplify the numerator:

3x - 24 - 2 = 3x - 26

Therefore:

y = (3x - 26)/2

Which can be written as:

y = (3/2)x - 13

Wait, let me check that again. Because (3x - 26)/2 is the same as (3/2)x - 13. Because 26 divided by 2 is 13. So yes, that's correct.

So, the slope m is 3/2 and the y-intercept b is -13. Therefore, the ordered pair (m, b) is (3/2, -13). Let me verify this result to make sure I didn't make a mistake.

Let's test the parameterization with a specific value of t. Let's pick t = 0. Then x = 8 + 2*0 = 8, y = -1 + 3*0 = -1. Plugging x = 8 into the equation y = (3/2)x -13:

y = (3/2)*8 -13 = 12 -13 = -1. Correct.

Another test: let t = 1. Then x = 8 + 2*1 = 10, y = -1 + 3*1 = 2. Plugging x =10 into the equation:

y = (3/2)*10 -13 = 15 -13 = 2. Correct.

One more test: t = 2. x = 8 + 4 =12, y = -1 +6 =5. Using the equation: (3/2)*12 =18, 18 -13=5. Correct.

Seems like the equation works. Therefore, the ordered pair (m, b) is (3/2, -13). So, I think that's the answer.
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