At a cafeteria, Mary orders two pieces of toast and a bagel, which comes out to $3.15. Gary orders a bagel and two muffins, which comes out to $4.85. Larry orders a piece of toast, two bagels, and three muffins, which comes out to $8.25. How many cents does one bagel cost?
Let t be the cost of one piece of toast, b be the cost of one bagel, and m be the cost of one muffin. We can set up the following system of equations:
2t + b = 315 (Mary's order)
b + 2m = 485 (Gary's order)
t + 2b + 3m = 825 (Larry's order)
We want to find the value of b.
From the first equation, we can solve for t:
2t + b = 315
2t = 315 - b
t = (315 - b)/2
Substituting this into the third equation:
(315 - b)/2 + 2b + 3m = 825
Multiplying both sides by 2 to get rid of the fraction:
315 - b + 4b + 6m = 1650
Simplifying:
3b + 6m = 1335
Dividing by 3:
b + 2m = 445
Now we have two equations in two variables:
2t + b = 315
b + 2m = 445
Solving for b in the second equation:
b = 445 - 2m
Substituting this into the first equation:
2t + (445 - 2m) = 315
Simplifying:
2t - 2m = -130
Dividing by 2:
t - m = -65
Now we can solve for m in terms of t:
m = t + 65
Substituting back into the second equation:
b + 2(t + 65) = 445
Solving for b:
b + 2t + 130 = 445
b = 315 - 2t
Substituting into the first equation:
2t + (315 - 2t) = 315
2t = 0
t = 0
Substituting t = 0 back into the equation for b:
b = 315 - 2t = 315
Therefore, one bagel costs 315 cents.
Let's assign variables to represent the cost of each item:
Let x represent the cost of one piece of toast,
Let y represent the cost of one bagel,
Let z represent the cost of one muffin.
From the given information, we can form the following equations:
Mary's order:
2x + 1y = 3.15 --(1)
Gary's order:
1y + 2z = 4.85 --(2)
Larry's order:
1x + 2y + 3z = 8.25 --(3)
We need to find the value of y, i.e., the cost of one bagel.
To solve these equations, we need to eliminate one variable from them.
We'll start by multiplying equation (1) by 2 to make the coefficients of 'y' equal in equation (3):
4x + 2y = 6.30 --(4)
Now, we can subtract equation (2) from equation (4) to eliminate 'y':
(4x + 2y) - (1y + 2z) = 6.30 - 4.85
4x + 2y - y - 2z = 6.30 - 4.85
4x + y - 2z = 1.45 --(5)
Next, we'll multiply equation (1) by 3 and equation (2) by 2 to make the coefficients of 'y' equal in equation (3):
6x + 3y = 9.45 --(6)
2y + 4z = 9.70 --(7)
Now, we can subtract equation (7) from equation (6) to eliminate 'y':
(6x + 3y) - (2y + 4z) = 9.45 - 9.70
6x + 3y - 2y - 4z = 9.45 - 9.70
6x + y - 4z = -0.25 --(8)
At this point, we have two equations (5) and (8) with the same coefficients for 'y':
4x + y - 2z = 1.45 --(5)
6x + y - 4z = -0.25 --(8)
Now, we can subtract equation (5) from equation (8) to eliminate 'y':
(6x + y - 4z) - (4x + y - 2z) = -0.25 - 1.45
6x + y - 4z - 4x - y + 2z = -0.25 - 1.45
2x - 2z = -1.70 --(9)
We now have two equations (5) and (9) without 'y':
4x + y - 2z = 1.45 --(5)
2x - 2z = -1.70 --(9)
To solve these two equations, let's multiply equation (9) by 2 to eliminate 'z':
(2x - 2z) * 2 = -1.70 * 2
4x - 4z = -3.40 --(10)
We can now add equation (10) to equation (5):
(4x + y - 2z) + (4x - 4z) = 1.45 - 3.40
4x + y - 2z + 4x - 4z = -1.95
8x + y - 6z = -1.95 --(11)
Now that we have one equation (11) with only x, y, and z, we can solve this system of equations.
Combining like terms, the equation becomes:
8x + y - 6z = -1.95
Substitute the values of x, y, and z from equations (1), (2), and (3) into equation (11):
8(0.6) + 0.6 - 6(0.9) = -1.95
4.8 + 0.6 - 5.4 = -1.95
5.4 - 5.4 = -1.95 - 0.6
0 = -2.55
There is no solution to this system of equations.
Thus, there is no unique value for the cost of one bagel. The given information is inconsistent, and we cannot determine the cost of one bagel.