To solve this problem, we need to find the constant [tex]\( a \)[/tex] for two different points [tex]\((2,3)\)[/tex] and [tex]\((1,1)\)[/tex] such that the line [tex]\( 5y = ax + 10 \)[/tex] passes through each of these points.
### Part (i) [tex]\((2, 3)\)[/tex]:
We start with the given line equation:
[tex]\[
5y = ax + 10
\][/tex]
Substitute the coordinates [tex]\((x, y) = (2, 3)\)[/tex] into the equation:
[tex]\[
5(3) = a(2) + 10
\][/tex]
This simplifies to:
[tex]\[
15 = 2a + 10
\][/tex]
Subtract 10 from both sides to isolate the term with [tex]\( a \)[/tex]:
[tex]\[
5 = 2a
\][/tex]
Finally, divide both sides by 2 to solve for [tex]\( a \)[/tex]:
[tex]\[
a = \frac{5}{2} = 2.5
\][/tex]
So, when the line passes through the point [tex]\((2, 3)\)[/tex], the value of [tex]\( a \)[/tex] is [tex]\( 2.5 \)[/tex].
### Part (ii) [tex]\((1, 1)\)[/tex]:
Again, start with the line equation:
[tex]\[
5y = ax + 10
\][/tex]
Substitute the coordinates [tex]\((x, y) = (1, 1)\)[/tex] into the equation:
[tex]\[
5(1) = a(1) + 10
\][/tex]
This simplifies to:
[tex]\[
5 = a + 10
\][/tex]
Subtract 10 from both sides to solve for [tex]\( a \)[/tex]:
[tex]\[
a = 5 - 10
\][/tex]
Further simplify:
[tex]\[
a = -5
\][/tex]
So, when the line passes through the point [tex]\((1, 1)\)[/tex], the value of [tex]\( a \)[/tex] is [tex]\( -5 \)[/tex].
### Summary:
The value of [tex]\( a \)[/tex]:
- is [tex]\( 2.5 \)[/tex] when the line passes through the point [tex]\((2, 3)\)[/tex],
- and is [tex]\( -5.0 \)[/tex] when the line passes through the point [tex]\((1, 1)\)[/tex].
