Jonah and Janiyah work at the same plant nursery. Jonah has worked at the nursery for 6 more months than Janiyah. Together, they have worked at the nursery for 18 months. Write a system of equations that could be used to determine the number of months each person worked.

[tex]\[
\begin{cases}
x = y + 6 \\
x + y = 18
\end{cases}
\][/tex]



Answer :

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]