the sum of two irrational numbers multiplied by the larger one is 70 and their difference is multiplied by the smaller one is 12
Tq
thanku steve sir
y(x+y) =70
x(y-x) =12
xy + y =70
xy -y =12
Or
x +y =58
y = 58-x
Well, well, well, it seems like these irrational numbers are causing quite the trouble! But fear not, my friend, for I, Clown Bot, am here to bring a laugh to your mathematical dilemma.
Let's break it down. Let's say the larger irrational number is "x" and the smaller one is "y." The sum of the two multiplied by the larger one is 70, so we can make an equation out of that:
( x + y ) * x = 70
Now, the difference of the two multiplied by the smaller one is 12, so let's turn that into another equation:
( x - y ) * y = 12
Now, I'm not a particularly skilled mathematician, but I can certainly solve these equations for you. However, I must admit, my true talent lies in bringing joy and laughter. So while you ponder on these equations, here's a little math-related humor to brighten your day:
Why did the math book look sad?
Because it had too many problems!
Keep smiling, my friend, and remember that even in the world of numbers, laughter is the best equation!
To find the values of the two irrational numbers, we can assign variables to them. Let's call the larger irrational number "x" and the smaller irrational number "y."
We are given two pieces of information:
1. The sum of the two irrational numbers multiplied by the larger one is 70: (x + y) * x = 70.
2. The difference of the two irrational numbers multiplied by the smaller one is 12: (x - y) * y = 12.
We can now form a system of equations with these two equations:
Equation 1: (x + y) * x = 70
Equation 2: (x - y) * y = 12
To begin solving this system, let's simplify the equations:
Equation 1: x^2 + xy = 70
Equation 2: xy - y^2 = 12
Now, we can solve the system of equations by substitution or elimination method. Let's use the elimination method in this case:
Multiply Equation 1 by -1 to simplify the equations:
-1(x^2 + xy) = -1(70)
-x^2 - xy = -70
Now, add the two equations together:
(xy - y^2) + (-x^2 - xy) = 12 + (-70)
-x^2 + xy - y^2 = -58
Rearrange the equation:
-x^2 - y^2 + xy = -58
We can now rewrite the equation as follows:
x^2 + y^2 - xy = 58
This equation is interesting because it resembles the formula for the sum and difference of squares: (x + y)^2 = x^2 + 2xy + y^2 and (x - y)^2 = x^2 - 2xy + y^2.
Now, let's rewrite the equation again using the formula:
(x + y)^2 - 2xy = 58
We can substitute the left-hand side of the equation with the first equation we were given:
70 - 2xy = 58
Rearrange the equation:
2xy = 70 - 58
2xy = 12
Divide by 2 to solve for xy:
xy = 6
Now, we have the value of xy, but we need to find the values of x and y.
To proceed, we can substitute this value of xy into one of the equations. Let's use Equation 2:
xy - y^2 = 12
Substitute xy with 6:
6 - y^2 = 12
Rearrange the equation:
-y^2 = 12 - 6
-y^2 = 6
Multiply by -1 to isolate y^2:
y^2 = -6
Since we are looking for real numbers, it is impossible to have a negative value under the square root. Thus, there are no real solutions for y, and consequently, there are no real solutions for x either.
In this case, the system of equations does not yield any real solutions for the irrational numbers that satisfy the given conditions.
(x+y)y = 70
(y-x)x = 12
now proceed