Let's solve the problems step-by-step:
### Part (a): Finding the value of [tex]\( y \)[/tex]
You are given that:
- [tex]\( RS = 9y + 2 \)[/tex]
- [tex]\( ST = 2y + 5 \)[/tex]
- [tex]\( RT = 51 \)[/tex]
Since [tex]\( R \)[/tex], [tex]\( S \)[/tex], and [tex]\( T \)[/tex] are points on a number line where [tex]\( RS \)[/tex] and [tex]\( ST \)[/tex] are segments between the points, the sum of [tex]\( RS \)[/tex] and [tex]\( ST \)[/tex] should be equal to [tex]\( RT \)[/tex]:
[tex]\[ RS + ST = RT \][/tex]
Substituting the given expressions:
[tex]\[ (9y + 2) + (2y + 5) = 51 \][/tex]
Combine like terms:
[tex]\[ 9y + 2y + 2 + 5 = 51 \][/tex]
[tex]\[ 11y + 7 = 51 \][/tex]
To isolate [tex]\( y \)[/tex], first subtract 7 from both sides:
[tex]\[ 11y = 44 \][/tex]
Then divide by 11:
[tex]\[ y = 4 \][/tex]
So, the value of [tex]\( y \)[/tex] is 4.
### Part (b): Finding RS and ST
Now that we know [tex]\( y = 4 \)[/tex], we can substitute this value back into the expressions for [tex]\( RS \)[/tex] and [tex]\( ST \)[/tex] to find their lengths.
For [tex]\( RS \)[/tex]:
[tex]\[ RS = 9y + 2 \][/tex]
[tex]\[ RS = 9(4) + 2 \][/tex]
[tex]\[ RS = 36 + 2 \][/tex]
[tex]\[ RS = 38 \][/tex]
For [tex]\( ST \)[/tex]:
[tex]\[ ST = 2y + 5 \][/tex]
[tex]\[ ST = 2(4) + 5 \][/tex]
[tex]\[ ST = 8 + 5 \][/tex]
[tex]\[ ST = 13 \][/tex]
### Summary
- The value of [tex]\( y \)[/tex] is 4.
- The length of segment [tex]\( RS \)[/tex] is 38.
- The length of segment [tex]\( ST \)[/tex] is 13.
