To find the value of [tex]\( y \)[/tex] for the point [tex]\((6, y)\)[/tex] that lies on the same line as the point [tex]\((10, -1)\)[/tex] with a given slope of [tex]\(\frac{1}{4}\)[/tex]:
1. Identify the coordinates and slope:
- First point: [tex]\((10, -1)\)[/tex]
- Second point: [tex]\((6, y)\)[/tex]
- Slope, [tex]\( m = \frac{1}{4} \)[/tex]
2. Use the slope formula:
The slope [tex]\(m\)[/tex] between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is given by:
[tex]\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
3. Substitute the known values:
Here, [tex]\( (x_1, y_1) = (10, -1) \)[/tex] and [tex]\( (x_2, y_2) = (6, y) \)[/tex]. Substituting these values into the slope formula, we get:
[tex]\[
\frac{1}{4} = \frac{y - (-1)}{6 - 10}
\][/tex]
4. Simplify the denominator:
Since [tex]\( 6 - 10 = -4 \)[/tex], the equation becomes:
[tex]\[
\frac{1}{4} = \frac{y + 1}{-4}
\][/tex]
5. Cross-multiply to solve for [tex]\( y \)[/tex]:
Cross-multiplying the terms gives:
[tex]\[
1 \cdot (-4) = 4 \cdot (y + 1)
\][/tex]
Simplifying this, we get:
[tex]\[
-4 = 4(y + 1)
\][/tex]
6. Solve for [tex]\( y \)[/tex]:
Divide both sides by 4:
[tex]\[
-1 = y + 1
\][/tex]
Subtract 1 from both sides to isolate [tex]\( y \)[/tex]:
[tex]\[
y = -2
\][/tex]
So, the value of [tex]\( y \)[/tex] is:
[tex]\[
-2
\][/tex]
Thus, the correct answer is:
[tex]\[ \boxed{-2} \][/tex]
