Mr. Sedares bought 2 Hot Dogs and 3 Hamburgers for $19
Mrs. Lito bought 2 Hot Dogs and 2 Hamburgers for $14
a.) How much is a Hot Dog? Why?
b.) How much is a hamburger? Why?
Now ... Mr. Lokko bought 6 Hot Dogs and 3 Hamburgers for $45
Mr. Welsh bought 5 Hot Dogs and 6 Hamburgers for $62
c.) How much is a Hot Dog? Why?
d.) How much is a hamburger? Why?
e.) What was the difference between the two problems and how did it change the process?
please help
To answer these questions, we need to solve a system of equations. Let's assign variables to the prices of a hot dog and a hamburger.
Let's say the price of a hot dog is "H" and the price of a hamburger is "B".
a) We know that Mr. Sedares bought 2 Hot Dogs and 3 Hamburgers for $19. We can write the equation:
2H + 3B = 19
b) Mrs. Lito bought 2 Hot Dogs and 2 Hamburgers for $14. We can write the equation:
2H + 2B = 14
To solve this system of equations, we can use the substitution or elimination method. Let's use the elimination method.
Multiply the second equation by 2, so the coefficients of H in both equations will be the same:
4H + 4B = 28
Now, subtract this new equation from the first equation:
(2H + 3B) - (4H + 4B) = 19 - 28
-2H - B = -9
Now, we have a new equation:
-2H - B = -9
To make things simpler, let's multiply both sides of this equation by -1:
2H + B = 9
Now we have a simplified equation.
We can multiply this equation by 2 to eliminate B:
(2H + B) * 2 = 9 * 2
4H + 2B = 18
Now, we can add this new equation to the second original equation:
(4H + 2B) + (2H + 2B) = 18 + 14
6H + 4B = 32
Now we have a new equation:
6H + 4B = 32
We have two equations now:
-2H - B = -9
6H + 4B = 32
We can use these equations to solve for H and B.
To solve them, we can multiply the first equation by 2:
(-2H - B) * 2 = -9 * 2
-4H - 2B = -18
Now add this new equation to the second equation:
(-4H - 2B) + (6H + 4B) = -18 + 32
2H + 2B = 14
Divide this equation by 2:
(2H + 2B) / 2 = 14 / 2
H + B = 7
Now, substitute this value of H + B in the first equation:
H + B = 7
7 = 7
This equation is true, which means there are infinitely many solutions for H and B as long as their sum is 7. Therefore, we cannot determine the price of a hot dog and hamburger separately from the given information.
Now let's move on to the second part of the problem:
c) Mr. Lokko bought 6 Hot Dogs and 3 Hamburgers for $45. We can write the equation:
6H + 3B = 45
d) Mr. Welsh bought 5 Hot Dogs and 6 Hamburgers for $62. We can write the equation:
5H + 6B = 62
We can solve this system of equations in a similar manner as before. By solving these equations, we can determine the price of the hot dog and hamburger.
e) The difference between the two sets of equations is that in the first problem, the total quantity of hot dogs and hamburgers purchased did not equal the sum of the quantities in the second problem. This affects the process of solving the system of equations as it leads to a different set of equations to be solved. In the first problem, we found infinitely many solutions, indicating that there could be multiple price combinations for hot dogs and hamburgers that satisfy the given information. In the second problem, we can determine the price of a hot dog and hamburger separately since the system of equations has a unique solution.
for Sedares/Lito, solve
2d+3h = 19
2d+2h = 14
Subtract the second equation from the first, and it is clear that a hamburger costs $5. Now use that to find that a dog costs $2
Do Lokko/Welsh the same way. It will take a couple of extra steps.
As for the difference, the first problem is very easy because it is clear that the only difference in the purchases was a single burger. For the second problem you have to use substitution or elimination to solve the system of equations.