Several customers order small fruit baskets filled with apples, bananas, and oranges. Let a represent the price per pound of apples, $b$ represent the price per pound of bananas, and $c$ represent the price per pound of oranges. The system represents the number of pounds of each type of fruit and the total price of each fruit basket. How much per pound does each type of fruit cost?
[tex]\left\{\begin{array}{r}
a+2 b+3 c=12 \\
3 a+2 b+5 c=22 \\
2 a+4 b+4 c=18
\end{array}\right.[/tex]
Answer (2)
The prices per pound of the fruits are: apples at $2.00, bananas at $0.50, and oranges at $3.00.
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Eliminate b from the first two equations to get a + c = 5 .
Eliminate b from the first and third equations to get 2 c = 6 , so c = 3 .
Substitute c = 3 into a + c = 5 to find a = 2 .
Substitute a = 2 and c = 3 into the first equation to find b = 0.5 .
The solution is a = 2 , b = 0.5 , c = 3 .
Explanation
Understanding the Problem We are given a system of three linear equations with three unknowns a, b, and c, representing the price per pound of apples, bananas, and oranges, respectively. Our goal is to find the values of a, b, and c.
Setting up the Solution The system of equations is:
{ a + 2 b + 3 c = 12 3 a + 2 b + 5 c = 22 2 a + 4 b + 4 c = 18
We can solve this system using elimination.
Eliminating b from Equations 1 and 2 First, let's eliminate b from the first two equations. Subtract the first equation from the second equation:
( 3 a + 2 b + 5 c ) − ( a + 2 b + 3 c ) = 22 − 12 , which simplifies to 2 a + 2 c = 10 . Dividing by 2, we get a + c = 5 .
Eliminating b from Equations 1 and 3 Next, let's eliminate b from the first and third equations. Multiply the first equation by 2: 2 ( a + 2 b + 3 c ) = 2 ( 12 ) , which gives 2 a + 4 b + 6 c = 24 . Subtract the third equation from this result: ( 2 a + 4 b + 6 c ) − ( 2 a + 4 b + 4 c ) = 24 − 18 , which simplifies to 2 c = 6 , so c = 3 .
Solving for a Now, substitute c = 3 into a + c = 5 to find a = 5 − 3 = 2 .
Solving for b Finally, substitute a = 2 and c = 3 into the first equation a + 2 b + 3 c = 12 to find 2 + 2 b + 3 ( 3 ) = 12 , which gives 2 b = 12 − 2 − 9 = 1 , so b = 0.5 .
Final Answer Therefore, the solution is a = 2 , b = 0.5 , and c = 3 . This means apples cost $2.00/ l b , bananas cost $0.50/ l b , and oranges cost $3.00/ l b .
Examples
Understanding systems of equations can help in various real-world scenarios. For instance, if you're running a small business selling different products, you can use a system of equations to determine the optimal pricing for each product to maximize your profit. By setting up equations that represent your costs, sales volume, and desired profit, you can solve for the prices that meet your business goals. This ensures you're not overpricing or underpricing your products, leading to better financial outcomes.
