x + y = 78 (since Rika has to hand out a total of 114 samples, and she has already handed out 36 samples, which leaves 78 samples to hand out)
x/3 = (78-y)/4 (since Rika must hand out _1
3 the number of fragrance samples and _1
4 the number of hand lotion samples she must hand out. So, x/3 represents the number of fragrance samples she must hand out and (78-y)/4 represents the number of hand lotion samples she must hand out. We set these two expressions equal to each other since they represent the same thing - the total number of samples she has to hand out.)
Solve the system using your chosen method
Let's use the substitution method to solve the system of equations. We can start by solving the first equation for x or y:
x = 78 - y
Now we can substitute this expression for x into the second equation:
(78 - y)/3 = (78 - y)/4
Now we can solve the equation for y. Start by multiplying both sides of the equation by 12 to get rid of the fractions:
4(78 - y) = 3(78 - y)
Expand:
312 - 4y = 234 - 3y
Now, add 4y to both sides and subtract 234 from both sides:
78 = y
Now that we have a value for y, we can substitute it back into our expression for x:
x = 78 - 78 = 0
So, Rika should hand out 0 more fragrance samples (x) and 78 more hand lotion samples (y).