To solve the system of equations:
[tex]\[
\begin{cases}
d = 18t + 34 \\
d = 22t
\end{cases}
\][/tex]
we can use substitution or elimination. Here, we'll use substitution methodology to find the values of [tex]\( d \)[/tex] and [tex]\( t \)[/tex] that make both equations true simultaneously.
### Step-by-Step Solution:
1. Set the equations equal to each other:
Since both expressions equal [tex]\( d \)[/tex], we can set them equal to each other:
[tex]\[
18t + 34 = 22t
\][/tex]
2. Solve for [tex]\( t \)[/tex]:
To isolate [tex]\( t \)[/tex], subtract [tex]\( 18t \)[/tex] from both sides of the equation:
[tex]\[
34 = 22t - 18t
\][/tex]
Simplify the right side:
[tex]\[
34 = 4t
\][/tex]
Now, divide both sides by 4 to solve for [tex]\( t \)[/tex]:
[tex]\[
t = \frac{34}{4} = 8.5
\][/tex]
3. Substitute [tex]\( t \)[/tex] back to find [tex]\( d \)[/tex]:
Use the value of [tex]\( t = 8.5 \)[/tex] in either of the original equations to find [tex]\( d \)[/tex]. We'll use [tex]\( d = 22t \)[/tex]:
[tex]\[
d = 22 \times 8.5 = 187
\][/tex]
So, the solution to the system of equations is:
[tex]\[
\boxed{(8.5, 187)}
\][/tex]
### Verification with Options:
Now let's compare this solution with the given options to find the correct one:
- Option A: [tex]\((5.5, 121)\)[/tex]
- Option B: [tex]\((8.5, 187)\)[/tex]
- Option C: [tex]\((17, 374)\)[/tex]
- Option D: [tex]\((34, 748)\)[/tex]
Comparing our result [tex]\((8.5, 187)\)[/tex] with the options, we see that:
[tex]\[
\boxed{B. (8.5, 187)}
\][/tex]
is the correct answer.
