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Zorluk: OrtaSystems of Linear Equations

At a local farmer's market, a vendor sells two types of fruit baskets: a standard basket and a deluxe basket. The standard basket contains 33 apples and 22 oranges. The deluxe basket contains 55 apples and 44 oranges. If a customer bought a combination of these baskets containing a total of 3131 apples and 2222 oranges, what is the total number of baskets the customer bought?

Cevap: 9

Cevap

The total number of baskets the customer bought is 9.
The correct answer is obtained by setting up a system of equations where ss is the number of standard baskets and dd is the number of deluxe baskets. The equations 3s+5d=313s + 5d = 31 and 2s+4d=222s + 4d = 22 represent the totals of apples and oranges respectively. Dividing the second equation by 2 gives s+2d=11s + 2d = 11, which can be rearranged to s=11−2ds = 11 - 2d. Substituting this expression into the first equation yields 3(11−2d)+5d=313(11 - 2d) + 5d = 31, which simplifies to 33−d=3133 - d = 31, meaning d=2d = 2. Substituting d=2d = 2 back gives s=7s = 7. The total number of baskets is the sum of both types, 7+2=97 + 2 = 9.

Adım Adım Çözüm

1
Define variables and set up the system of linear equations.
Let ss be the number of standard baskets and dd be the number of deluxe baskets. The equations are: 3s+5d=313s + 5d = 31 (for apples) and 2s+4d=222s + 4d = 22 (for oranges).
To represent the relationships between the number of baskets and the total quantities of fruits mathematically.
2
Solve the system of equations using substitution.
From the second equation, dividing by 2 yields s+2d=11s + 2d = 11, so s=11−2ds = 11 - 2d. Substituting this into the first equation gives 3(11−2d)+5d=313(11 - 2d) + 5d = 31, which simplifies to 33−d=3133 - d = 31, leading to d=2d = 2.
To find the number of deluxe baskets purchased.
3
Calculate the number of standard baskets and sum the two counts to find the total number of baskets.
Substitute d=2d = 2 into s=11−2ds = 11 - 2d to get s=7s = 7. The total number of baskets is s+d=7+2=9s + d = 7 + 2 = 9.
To answer the specific question asking for the total number of baskets bought.

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Systems of Linear Equations
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Systems of Linear Equations

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