To determine the value of [tex]\( y \)[/tex] for which the line passing through the points [tex]\((7,7)\)[/tex] and [tex]\((4,y)\)[/tex] is perpendicular to the line [tex]\( 3x + 10y = 6 \)[/tex], we can follow these steps:
1. Determine the slope of the given line [tex]\( 3x + 10y = 6 \)[/tex]:
- Convert the equation to the slope-intercept form [tex]\( y = mx + b \)[/tex].
- Start with [tex]\( 3x + 10y = 6 \)[/tex].
- Solve for [tex]\( y \)[/tex]:
[tex]\[
10y = -3x + 6
\][/tex]
[tex]\[
y = -\frac{3}{10}x + \frac{6}{10}
\][/tex]
- Therefore, the slope [tex]\( m \)[/tex] of the given line is [tex]\( -\frac{3}{10} \)[/tex].
2. Determine the slope of the line that is perpendicular to this line:
- The slopes of two perpendicular lines are negative reciprocals of each other. This means if the slope of one line is [tex]\( m \)[/tex], the slope of the line perpendicular to it is [tex]\( -\frac{1}{m} \)[/tex].
- Therefore, the slope of the line through [tex]\((7,7)\)[/tex] and [tex]\((4,y)\)[/tex] is:
[tex]\[
-\frac{1}{-\frac{3}{10}} = \frac{10}{3}
\][/tex]
3. Use the slope formula to find [tex]\( y \)[/tex]:
- The slope formula for the line passing through points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is:
[tex]\[
\text{slope} = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
- Substitute [tex]\( x_1 = 7 \)[/tex], [tex]\( y_1 = 7 \)[/tex], [tex]\( x_2 = 4 \)[/tex], [tex]\( y_2 = y \)[/tex], and the slope [tex]\( \frac{10}{3} \)[/tex]:
[tex]\[
\frac{y - 7}{4 - 7} = \frac{10}{3}
\][/tex]
[tex]\[
\frac{y - 7}{-3} = \frac{10}{3}
\][/tex]
- Solve for [tex]\( y \)[/tex]:
[tex]\[
y - 7 = -3 \times \frac{10}{3}
\][/tex]
[tex]\[
y - 7 = -10
\][/tex]
[tex]\[
y = -10 + 7
\][/tex]
[tex]\[
y = -3
\][/tex]
Therefore, the value of [tex]\( y \)[/tex] is [tex]\( -3 \)[/tex]. Thus, the correct answer is:
[tex]\[ y = -3 \][/tex]
So the answer is [tex]\( \boxed{-3} \)[/tex].
