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2 years ago

What is the solution to the system of equations below?

Mathematics

1 answer:

Lelechka [254]2 years ago

6 0

Answer:

Step-by-step explanation:

For x+3y=15

Take (6,3) and substitute the points:

6+3(3)=15

6+9=15

15=15

For 4 x + 2 y = 30 and substitute the points:

4(6)+2(3)=30

24+6=30

30=30

So, (6,3) is the solution of the given system of equations.

<h3>If you need to ask any questions, let me know.</h3>

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The time value of money calculation can be performed using formula equations or online calculators.

The correct responses are;

1) Option 3

2) Option 2

3) The difference in principal is approximately $8,000

The difference in interest earned is approximately $2,977.87

4) It is better to invest more money at the beginning of the 30 years

Reasons:

Option 1: Present value = 0

Amount invested per month, A = $25/month

The Annual Percentage Rate, APR, r = 3.25%

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$\displaystyle FV_{A} = \mathbf{A \cdot \left (\frac{ \left(1 + \frac{r}{m} \right)^{m\cdot t} - 1}{\frac{r}{m} } \right)}$

In option 1, m = 12 periods per year

Therefore;

$\displaystyle FV_{A} = 25 \times \left (\frac{ \left(1 + \frac{0.0325}{12} \right)^{12 \times 30} - 1}{\frac{0.0325}{12} } \right) \approx \mathbf{15,209.3}$

Contribution = $25 × 12 × 30 = $9,000

Total interest earned = $15,209.3 - $9,000 = $6,209.3

Final balance = $15,209.3

Option 2: Present value = 0

Amount, A = $75/quarter

m = 4 periods per year

The Annual Percentage Rate, APR = 4.00%

Therefore;

The effective interest rate is therefore;

$\displaystyle r_{eff} = \left(1 + \frac{0.04}{4} \right)^4 - 1 \approx \mathbf{0.04060401}$

$\displaystyle FV_{A} = 75 \times \left (\frac{ \left(1 + \frac{0.04060401}{4} \right)^{4 \times 30} - 1}{\frac{0.04060401}{4} } \right) \approx 17,437.7$

Using an online calculator, FV = $17,467.04

Contribution = $75 × 4 × 30 = $9,000

Total interest earned = $17,467.04 - $9,000 = $8,467.04

Final balance = $17,467.04

Option 3: Present value = $1,000

APR = 6.25%

m = 12 period per year

Number of years, t = 30 years

Therefore;

$\displaystyle FV = \left (1 + \frac{0.0625}{12} \right)^{12 \times 30} \approx \mathbf{6,489.17}$

Contribution = $1,000

Total interest earned = $6,489.17 - $1,000 = $5,489.17

Final balance = $6,489.17

The table of values is therefore;

$\begin{tabular}{|c|c|c|c|}Option \# &Contribution &Total Interest Earned&Final Balance\\1&\$9,000&\$6,209.3 & \$15,209.3\\2&\$9,000&\$8,467.04 &\$17,467.04\\3&\$1,000&\$5,489.17&\$6,489.17\end{array}\right]$

1) The option that has the least amount invested are <u>option 3</u>

Option 3 investment plan is a present value of $1,000, invested for 30 years at 6.25% APR compounded monthly.

2) <u>Option 2</u> yielded the highest amount at the end of 30 years, given that the APR is higher than the APR for option 1, although the amount invested over the period are the same.

The basis of option 2 investment plan is $75 invested quarterly at 4.00% APR compounded monthly for 30 years.

3) The difference in the principal invested for the highest and lowest final balance is $9,000 - $1,000 = <u>$8,000</u>

The difference in the interest earned is; $8,467.04 - $5,489.17 = <u>$2,977.87</u>

4) In option 1 the present value is zero, therefore zero amount was invested at the beginning.

The interest to investment ration is 6,209.3:9,000 ≈ 0.7:1

In option 3, all the money was invested at the beginning.

The interest to investment ratio of option 3 is; 5,489.17:1,000 ≈ 5.5:1

Given that the interest to investment ratio, which is the return on investment is larger when more money is saved at the beginning as in option 3, <u>it is better to invest more money at the beginning</u>.

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