Certainly! Let's solve the given system of equations step-by-step to find when the two investments will have the same value.
The equations we have are:
1. [tex]\( A = 54 + 5t \)[/tex]
2. [tex]\( A = 84 + 2t \)[/tex]
We assume that both equations represent the value [tex]\( A \)[/tex] at the same point in time [tex]\( t \)[/tex]. Thus, we'll set the right-hand sides of the two equations equal to each other and solve for [tex]\( t \)[/tex].
### Step-by-Step Solution
1. Set the equations equal to each other:
[tex]\[
54 + 5t = 84 + 2t
\][/tex]
2. Get all terms involving [tex]\( t \)[/tex] on one side of the equation:
Subtract [tex]\( 2t \)[/tex] from both sides:
[tex]\[
54 + 5t - 2t = 84
\][/tex]
Simplify the equation:
[tex]\[
54 + 3t = 84
\][/tex]
3. Isolate the term with [tex]\( t \)[/tex]:
Subtract 54 from both sides:
[tex]\[
3t = 84 - 54
\][/tex]
Simplify the right-hand side:
[tex]\[
3t = 30
\][/tex]
4. Solve for [tex]\( t \)[/tex]:
Divide both sides by 3:
[tex]\[
t = \frac{30}{3}
\][/tex]
Simplify:
[tex]\[
t = 10
\][/tex]
### Interpretation
The solution [tex]\( t = 10 \)[/tex] means that both investments will have the same value after 10 years.
Thus, the correct interpretation is:
d. [tex]\( t = 10 \)[/tex]
The two investments will reach the same value in 10 years.
This is the accurate conclusion drawn from solving the given system of equations.
