Question 1203103: A customer bought 2 fruit cups and 3 sandwiches for US$ 19.
When she bought 6 fruit cups and 8 sandwiches her bill was US $54.
• Write two equations to represent the information given above.
• Calculate the cost for ONE fruit cup.
• Hence, determine the cost per sandwich.
Your friend, Manuel, got this question on a take home assignment. Having solved the pair of equations, he believes that the cost per fruit cup is US$3 and the cost per sandwich is US $5.
a) Write the pair of equations, clearly defining each variable used. [1 mark]
b) Use a graphical method to tell whether Manuel is correct. You must clearly label the solution and explicitly state your position on Manuel’s answer. [2 marks]
c) Show Manuel an alternative approach on your College Algebra syllabus for solving pairs of simultaneous equations. You should name the method you will use and show full working. [3 marks]
Found 2 solutions by math_tutor2020, josgarithmetic:
Answer by math_tutor2020(3817)   (Show Source):
You can put this solution on YOUR website!
Part (a)
x = cost of 1 fruit cup
y = cost of 1 sandwich
Each cost is in dollars.
2x = cost of 2 fruit cups
3y = cost of 3 sandwiches
2x+3y = cost of 2 fruit cups and 3 sandwiches
2x+3y = 19
Through a similar thought process, the other equation would be 6x+8y = 54
The system of equations is
2x+3y = 19
6x+8y = 54
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Part (b)
Use GeoGebra, Desmos, TI83/84, or any graphing tool to plot the equations we found in the previous part.
Another approach is to plot the equations by hand on a piece of graph paper.
2x+3y = 19 goes through the points (2,5) and (5,3)
6x+8y = 54 goes through the points (1,6) and (5,3)
The two lines intersect at the point (5,3)
It is the single solution to the system.
This breaks down to x = 5 and y = 3
Therefore, 1 fruit cup costs $5 and 1 sandwich costs $3
Manuel has the two values incorrectly swapped.
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Part (c)
We can use the elimination method as one approach.
Triple each side of the 1st equation
2x+3y = 19
3(2x+3y) = 3*19
6x+9y = 57
This original system
2x+3y = 19
6x+8y = 54
is equivalent to this system
6x+9y = 57
6x+8y = 54
From here, subtract straight down.
The x terms cancel out because 6x-6x = 0x = 0
The y terms turn into 9y-8y = 1y = y
The right-hand-sides become 57-54 = 3
Ultimately this boils down to y = 3.
Use this y value to find x.
2x+3y = 19
2x+3*3 = 19
2x+9 = 19
2x = 19-9
2x = 10
x = 10/2
x = 5
Or we could pick on the other equation
6x+8y = 54
6x+8*3 = 54
6x+24 = 54
6x = 54-24
6x = 30
x = 30/6
x = 5
Either way, we have determined that x = 5
So that's how we can back up part (b) when we mentioned x = 5 and y = 3.
Check:
Plug x = 5 and y = 3 into the 1st equation
2y+3y = 19
2*5+3*3 = 19
10+9 = 19
19 = 19
Repeat for the other equation
6x+8y = 54
6*5+8*3 = 54
30+24 = 54
54 = 54
Both equations are true for this pair of x,y values. The answer is fully confirmed.
Answer by josgarithmetic(39618) (Show Source):
You can put this solution on YOUR website! The first section:
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A customer bought 2 fruit cups and 3 sandwiches for US$ 19.
When she bought 6 fruit cups and 8 sandwiches her bill was US $54.
• Write two equations to represent the information given above.
• Calculate the cost for ONE fruit cup.
• Hence, determine the cost per sandwich.
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c fruit cup price
w sandwich price
Description part gives this system:
The next two bullet-points are straightforward.
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