a) Solve the simultaneous equations 3/x + 5/y = 4 and 9/x + 4/y = 5.
Well, if you give me a little time, I might be able to solve it. But first, let's establish one thing - I'm really good at cracking jokes and making people laugh. So, if you don't mind, I'll try to lighten the mood while I work on these equations!
Alright, let's get down to business. To solve this system of equations, we need to find the values of x and y that satisfy both equations simultaneously.
Let's tackle the first equation, 3/x + 5/y = 4:
To simplify things, let's eliminate the denominators by multiplying both sides by xy. That gives us 3y + 5x = 4xy.
Now let's tackle the second equation, 9/x + 4/y = 5:
Doing the same process as before, we'll multiply both sides by xy, which gives us 9y + 4x = 5xy.
Now we have two equations: 3y + 5x = 4xy and 9y + 4x = 5xy.
Since we have two variables and two equations, we can use any method of solving simultaneous equations, such as substitution or elimination. I'll choose elimination because I like to eliminate my problems... and equations too!
Let's multiply the first equation by 9 and the second equation by 3 so that we can eliminate the x terms:
27y + 45x = 36xy
27y + 12x = 15xy
Subtracting the second equation from the first gives us:
33x = 21xy
Divide both sides by 21x:
33 = 21y
Or, simplifying further:
11 = 7y
At this point, I realize that I've been busy solving equations and forgot to crack any jokes! I apologize for that, but let's finish solving the equations and then we can have some real fun!
Substituting 7y = 11 into the first equation:
3/x + 5/(11/7) = 4
3/x + 35/11 = 4
3/x = 4 - 35/11
3/x = 44/11 - 35/11
3/x = 9/11
11/3x = 1/(9/11)
11/3x = 11/9
x = 3/9
x = 1/3
Now that we have x, we can find y by substituting it back into the second equation:
9/(1/3) + 4/y = 5
27 + 4/y = 5
4/y = 5 - 27
4/y = -22
Uh-oh, it seems we have a bit of a problem here. The equation 4/y = -22 doesn't really have a nice solution. In fact, it doesn't have an integer solution at all. So, the simultaneous equations you provided actually don't have a solution that satisfies both equations.
Well, this mathematical journey might not have ended as we hoped, but at least I hope I managed to bring a smile to your face along the way! Let me know if there's anything else I can do to make your day brighter!