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]$$54$[/tex] at [tex]$9.26 \%$[/tex] for [tex]$t$[/tex] years is given by
[tex]$A=54+5 t$[/tex]
The equation to model the value, [tex]$A$[/tex], of an investment of [tex]$$84$[/tex] at [tex]$2.38 \%$[/tex] 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]
The two investments will reach the same value in 5 years.
b. [tex]$t=20$[/tex]
The two investments will reach the same value in 20 years.
c. [tex]$t =1000$[/tex]
The two investments will reach the same value in 1000 years.
d. [tex]$t =10$[/tex]
The two investments will reach the same value in 10 years.
Answer (2)
Set the two equations equal to each other: 54 + 5 t = 84 + 2 t .
Simplify the equation by subtracting 2 t and 54 from both sides: 3 t = 30 .
Solve for t by dividing both sides by 3: t = 10 .
The two investments will reach the same value in 10 years: 10 .
Explanation
Setting up the Equations We are given two equations representing the value of two investments after t years:
A = 54 + 5 t A = 84 + 2 t
We need to find the value of t for which the two investments have the same value, A . This means we need to solve this system of two linear equations.
Equating the Expressions Since both equations are equal to A , we can set them equal to each other:
54 + 5 t = 84 + 2 t
Isolating the t term Now, we solve for t . First, subtract 2 t from both sides of the equation:
54 + 5 t − 2 t = 84 + 2 t − 2 t 54 + 3 t = 84
Further Isolating t Next, subtract 54 from both sides of the equation:
54 + 3 t − 54 = 84 − 54 3 t = 30
Solving for t Finally, divide both sides by 3 to find the value of t :
3 3 t = 3 30 t = 10
Interpreting the Result The two investments will reach the same value in 10 years. Therefore, the correct answer is:
d. t = 10 The two investments will reach the same value in 10 years.
Examples
Imagine you're saving money using two different plans. One plan starts with $54 and adds $5 each year. The other starts with $84 but only adds $2 each year. By setting up and solving a system of equations, we can determine how many years it will take for both plans to have the same amount of money. This type of problem is useful in comparing different investment options or savings plans to see when they might yield the same return.
The two investments will reach the same value in 10 years. This was determined by setting the equations equal to each other and solving for t , yielding t = 10 . The correct answer is d. t = 10 .
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