To find the value of [tex]\( n \)[/tex] where the line passes through the points [tex]\((3, n)\)[/tex] and [tex]\((6, 1)\)[/tex] and has a gradient (slope) of [tex]\(\frac{4}{5}\)[/tex], we use the formula for the gradient between two points:
[tex]\[
\text{Gradient} = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
Given:
- Points [tex]\((3, n)\)[/tex] and [tex]\((6, 1)\)[/tex]
- Gradient = [tex]\(\frac{4}{5}\)[/tex]
Substitute the given points into the gradient formula:
[tex]\[
\frac{1 - n}{6 - 3} = \frac{4}{5}
\][/tex]
Simplify the denominator:
[tex]\[
\frac{1 - n}{3} = \frac{4}{5}
\][/tex]
To clear the fraction, cross-multiply:
[tex]\[
5 \cdot (1 - n) = 4 \cdot 3
\][/tex]
Simplify both sides:
[tex]\[
5 - 5n = 12
\][/tex]
Isolate [tex]\( n \)[/tex] by first moving the constant term to the other side:
[tex]\[
-5n = 12 - 5
\][/tex]
[tex]\[
-5n = 7
\][/tex]
Divide both sides by [tex]\(-5\)[/tex]:
[tex]\[
n = \frac{7}{-5}
\][/tex]
[tex]\[
n = -\frac{7}{5}
\][/tex]
Thus, the value of [tex]\( n \)[/tex] is [tex]\(-\frac{7}{5}\)[/tex] which can also be written as [tex]\(-1.4\)[/tex]. However, since the problem specifies to give the answer as either an integer or a fraction in its simplest form, we keep it as:
[tex]\[
n = -\frac{7}{5}
\][/tex]
