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The formula [tex]\( A = P + P r t \)[/tex] represents the value, [tex]\( A \)[/tex], of an investment of [tex]\( P \)[/tex] dollars at a yearly simple interest rate, [tex]\( r \)[/tex], for [tex]\( t \)[/tex] years. The equation to model the value, [tex]\( A \)[/tex], of an investment of [tex]$54 at 9.26% for \( t \) years is given by:

\[ A = 54 + 5 t \]

The equation to model the value, \( A \), of an investment of $[/tex]84 at 2.38% for [tex]\( t \)[/tex] years is given by:

[tex]\[ A = 84 + 2 t \][/tex]

Assuming [tex]\( A \)[/tex] has the same value, the given equations form a system of two linear equations. Solve this system using an algebraic approach and interpret your answer.

a. [tex]\( t = 5 \)[/tex]
b. [tex]\( t = 20 \)[/tex]
c. [tex]\( t = 1000 \)[/tex]
d. [tex]\( t = 10 \)[/tex]

The two investments will reach the same value in:

A. 5 years
B. 20 years
C. 1000 years
D. 10 years



Answer :

GinnyAnswer
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.