Let's tackle the problem step-by-step.
### Given:
1. [tex]\( RS = 8y + 2 \)[/tex]
2. [tex]\( ST = 4y + 8 \)[/tex]
3. [tex]\( RT = 70 \)[/tex]
### Part a: What is the value of [tex]\( y \)[/tex]?
Since [tex]\( R \)[/tex], [tex]\( S \)[/tex], and [tex]\( T \)[/tex] are collinear points on a number line, we can write:
[tex]\[ RS + ST = RT \][/tex]
Substitute the given expressions for [tex]\( RS \)[/tex] and [tex]\( ST \)[/tex] into this equation:
[tex]\[ (8y + 2) + (4y + 8) = 70 \][/tex]
Combine like terms:
[tex]\[ 12y + 10 = 70 \][/tex]
To isolate [tex]\( y \)[/tex], first subtract 10 from both sides:
[tex]\[ 12y = 60 \][/tex]
Next, divide both sides by 12:
[tex]\[ y = 5 \][/tex]
### Part b: Find [tex]\( RS \)[/tex] and [tex]\( ST \)[/tex]
Now that we know [tex]\( y = 5 \)[/tex], substitute this value back into the expressions for [tex]\( RS \)[/tex] and [tex]\( ST \)[/tex]:
1. For [tex]\( RS \)[/tex]:
[tex]\[ RS = 8y + 2 \][/tex]
[tex]\[ RS = 8(5) + 2 \][/tex]
[tex]\[ RS = 40 + 2 \][/tex]
[tex]\[ RS = 42 \][/tex]
2. For [tex]\( ST \)[/tex]:
[tex]\[ ST = 4y + 8 \][/tex]
[tex]\[ ST = 4(5) + 8 \][/tex]
[tex]\[ ST = 20 + 8 \][/tex]
[tex]\[ ST = 28 \][/tex]
### Summary:
a. The value of [tex]\( y \)[/tex] is:
[tex]\[ y = 5 \][/tex]
b. The values of [tex]\( RS \)[/tex] and [tex]\( ST \)[/tex] are:
[tex]\[ RS = 42 \][/tex]
[tex]\[ ST = 28 \][/tex]
Therefore, the value of [tex]\( y \)[/tex] is 5.
