Let's solve this word problem step by step.
1. Identify Variables:
- Let [tex]\( x \)[/tex] be the number of months Jonah has worked.
- Let [tex]\( y \)[/tex] be the number of months Janiyah has worked.
2. Write Down the Relationships Given:
- Jonah has worked 6 more months than Janiyah: [tex]\( x = y + 6 \)[/tex].
- Together, they have worked for 18 months: [tex]\( x + y = 18 \)[/tex].
3. Form the System of Equations:
We have two key relationships here. These can be written as:
[tex]\[
\begin{cases}
x = y + 6 \\
x + y = 18
\end{cases}
\][/tex]
Now let's check which of the given systems matches this formulation:
- [tex]\( x > y + 6 \)[/tex]
- [tex]\( x + y < 18 \)[/tex]
- [tex]\( y < 6 \)[/tex]
- [tex]\( x + y > 18 \)[/tex]
- [tex]\( \begin{array}{l} y = 6 \\ x + y = 18 \end{array} \)[/tex]
- [tex]\( \begin{array}{l} x = y + 5 \\ x + y = 18 \end{array} \)[/tex]
The correct system among the choices is clearly:
[tex]\[
\begin{cases}
x = y + 6 \\
x + y = 18
\end{cases}
\][/tex]
4. Solve the System of Equations:
To find [tex]\( x \)[/tex] and [tex]\( y \)[/tex], we can substitute [tex]\( x = y + 6 \)[/tex] into the second equation:
[tex]\[
(y + 6) + y = 18
\][/tex]
Simplifying this, we get:
[tex]\[
2y + 6 = 18
\][/tex]
Subtract 6 from both sides:
[tex]\[
2y = 12
\][/tex]
Divide by 2:
[tex]\[
y = 6
\][/tex]
Now, substitute [tex]\( y = 6 \)[/tex] back into the equation [tex]\( x = y + 6 \)[/tex]:
[tex]\[
x = 6 + 6 = 12
\][/tex]
Therefore, Jonah has worked for 12 months and Janiyah has worked for 6 months. This matches the results.
Given the options, the system that could be used to determine the number of months each person worked is:
[tex]\[
\begin{cases}
x = y + 6 \\
x + y = 18
\end{cases}
\][/tex]
