Examkin

Soru

Zorluk: ZorSystems of Linear Equations

An investor allocates a total of $24,000\$24,000 among three accounts: Account A, which earns 3%3\% annual simple interest; Account B, which earns 5%5\% annual simple interest; and Account C, which earns 7%7\% annual simple interest. The total annual interest earned from all three accounts combined at the end of one year is $1,260\$1,260. If the amount invested in Account C is $2,000\$2,000 more than twice the amount invested in Account A, what is the amount invested in Account B?

  1. A
    $1,000\$1,000
  2. B
    $4,000\$4,000
  3. C
    $11,000\$11,000
  4. D
    $15,000\$15,000
  5. $19,000\$19,000Cevap

Cevap

The amount invested in Account B is $19,000\$19,000.
The system of linear equations representing the total investment, interest earned, and relative account values yields A=$1,000A = \$1,000, C=$4,000C = \$4,000, and B=$19,000B = \$19,000. Thus, the amount invested in Account B is $19,000\$19,000.

Adım Adım Çözüm

1
Define variables and write the system of three linear equations based on total investment, total annual interest, and account relationship.
Let AA, BB, and CC represent the dollars invested in Accounts A, B, and C respectively.
Equation (1): A+B+C=24,000A + B + C = 24,000
Equation (2): 0.03A+0.05B+0.07C=1,2600.03A + 0.05B + 0.07C = 1,260, which simplifies to 3A+5B+7C=126,0003A + 5B + 7C = 126,000
Equation (3): C=2A+2,000C = 2A + 2,000, or 2A−C=−2,0002A - C = -2,000
Translate the verbal conditions into an explicit 3×33 \times 3 system of linear equations.
2
Substitute C=2A+2,000C = 2A + 2,000 into Equations (1) and (2) to reduce the system to two variables (AA and BB).
From Equation (1): A+B+(2A+2,000)=24,000  ⟹  3A+B=22,000  ⟹  B=22,000−3AA + B + (2A + 2,000) = 24,000 \implies 3A + B = 22,000 \implies B = 22,000 - 3A.
From Equation (2): 3A+5B+7(2A+2,000)=126,000  ⟹  17A+5B+14,000=126,000  ⟹  17A+5B=112,0003A + 5B + 7(2A + 2,000) = 126,000 \implies 17A + 5B + 14,000 = 126,000 \implies 17A + 5B = 112,000.
Eliminate variable CC to simplify solving the linear system.
3
Substitute B=22,000−3AB = 22,000 - 3A into 17A+5B=112,00017A + 5B = 112,000 to solve for AA.
17A+5(22,000−3A)=112,000  ⟹  17A+110,000−15A=112,000  ⟹  2A=2,000  ⟹  A=1,00017A + 5(22,000 - 3A) = 112,000 \implies 17A + 110,000 - 15A = 112,000 \implies 2A = 2,000 \implies A = 1,000.
Solve for the single variable AA.
4
Determine the values of CC and BB.
C=2(1,000)+2,000=4,000C = 2(1,000) + 2,000 = 4,000.
B=22,000−3(1,000)=19,000B = 22,000 - 3(1,000) = 19,000.
Substitute A=1,000A = 1,000 back into the expressions for CC and BB to find the targeted investment amount.

Anahtar Kavram

Setting up and solving a system of three linear equations in three variables by substitution and elimination.
Tahmini Süre:2m 0s
Bu soruyu puanla

Konular

Systems of Linear Equations

Bu konudan daha fazla soru