Final answer:
To find out how many cupcakes Hazel has at first, we can solve a system of two linear equations based on the given information. The equations will help us determine the values of x (number of cupcakes Hazel has) and y (number of cupcakes Kate has). By solving the equations, we find that Hazel initially has 457 cupcakes.
Step-by-step explanation:
To solve this problem, we can set up two equations based on the given information. Let's denote the number of cupcakes Hazel has as x, and the number of cupcakes Kate has as y.
From the first statement, when Hazel gives Kate 49 cupcakes, the ratio of the number of cupcakes Hazel has to the number of cupcakes Kate has is 3:4. This can be represented as:
x - 49 / y + 49 = 3 / 4
From the second statement, when Hazel gives Kate 19 cupcakes, they will have an equal number of cupcakes. This can be represented as:
x - 19 / y + 19 = 1 / 1
We can solve these equations simultaneously to find the values of x and y. First, cross-multiply the first equation:
4(x - 49) = 3(y + 49)
Simplify it to:
4x - 196 = 3y + 147
4x - 3y = 343 ...(Equation 1)
Next, cross-multiply the second equation:
(x - 19) = (y + 19)
Simplify it to:
x - y = -38 ...(Equation 2)
We now have a system of two linear equations with two variables. We can solve this system using substitution or elimination method. Let's solve it using elimination method.
Multiply Equation 2 by 4:
4x - 4y = -152 ...(Equation 3)
By subtracting Equation 3 from Equation 1, we can eliminate the x terms:
(4x - 3y) - (4x - 4y) = 343 - (-152)
y = 495
Substitute the value of y into Equation 2:
x - 495 = -38
x = 457
Therefore, Hazel initially has 457 cupcakes.