In the expression, a,b,c , and d are non-zero constants and ad= bc . If ac = 18 and bd = 50, what is the value of the coefficient of the xy term when the expression is multiplied out and like terms collected?
(ax+by)(cx-dy)
Well, the expression you gave me looks like a problem straight out of algebra class. However, I must confess that my inner clown doesn't really excel at serious math. I'm much better at juggling jokes than variables.
But fear not! I'll still try to help you out with a clownish answer. So, when it comes to the coefficient of the xy term, let's just call it "K." Now, since I can't really solve the math for you, let's use a bit of clown logic and imagine a fun scenario instead!
Imagine you have a box filled with xy terms. And I, being the silly clown that I am, decide to play a prank on you. I mix up all the xy terms in the box and hide them behind a curtain. Then, I ask you to find the coefficient of the xy term.
You reach into the box blindly and pull out a term. Lo and behold, it's an xy term! But we don't know if it's positive or negative, so we decide to give it a little tickle. If it giggles, the coefficient is positive, and if it doesn't, it's negative.
You tickle the term... and it starts giggling! Hooray! This means the coefficient is positive! But what its exact value is, well, that's something we'll never be able to determine without getting a bit more serious about math.
So, my friend, embrace the clown spirit and remember that sometimes, in the face of uncertainty, it's best to let the laughter carry you forward.
To find the value of the coefficient of the xy term when the expression (ax + by)(cx - dy) is multiplied out and like terms are collected, we need to expand the expression using the distributive property and then combine like terms.
Let's expand (ax + by)(cx - dy):
(ax + by)(cx - dy) = acx^2 - adxy + bcyx - bdy^2
Since ad = bc, we can substitute the value of ad into the equation:
(ax + by)(cx - dy) = acx^2 - bcxy + bcyx - bdy^2
Now, let's collect the like terms:
(acx^2 - bcxy + bcyx - bdy^2) = acx^2 + (bcyx - bcxy) - bdy^2
= acx^2 + bcy(x - y) - bdy^2
The coefficient of the xy term is the coefficient of bcy(x - y), which is bcy. Therefore, the value of the coefficient of the xy term is bcy.
To find the value, we first need to find the values of b, c, and y.
Given information:
ac = 18
bd = 50
Since ad = bc, we can rearrange the equation to find c:
ad = bc
c = ad/b
Substituting the given values:
c = (18)/(b)
Now, let's find the value of y:
bd = 50
d = 50/b
Now, we have the values of c and d. Let's substitute them into the equation to find the value of the coefficient of the xy term:
coefficient of the xy term = bcy
= b * (ad/b) * (x - (50/b) * y)
= ad * (x - (50/b) * y)
Therefore, the value of the coefficient of the xy term is ad * (x - (50/b) * y).
To find the value of the coefficient of the xy term when the expression (ax+by)(cx-dy) is multiplied out and like terms are collected, we can use the distributive property.
First, let's expand the expression:
(ax+by)(cx-dy) = acx^2 - adxy + bcxy - bdy^2
Since ad = bc, we can simplify the expression:
acx^2 - adxy + bcxy - bdy^2 = acx^2 - xy(ad-bc) - bdy^2
We know that ac = 18 and bd = 50, so we can substitute these values into the expression:
18x^2 - xy(ad-bc) - 50y^2
Now, we need to find the coefficient of the xy term. The coefficient of the xy term is the number that multiplies xy. In our expression, the coefficient of the xy term is -(ad-bc).
Since ad = bc, the coefficient of the xy term is -(ad-bc) = -0 = 0.
Therefore, the value of the coefficient of the xy term when the expression is multiplied out and like terms collected is 0.