Solve simultaneously for the two equations for x and y
5x(cos 45) + 15y(cos 30) = 2,000
5x(sin45) - 15y(sin 30) = 0
I tried adding these two eqn. to elimate y and solve for x by adding [5 x cos 45] and [5 x sin 45] but the value for x is too high. Would I have to use trig identities to solve this equation for x and y?
Divide the first by 5, and replace the trig values
√2/2 x + 3√3/2 y = 400
times 2
√2x + 3√3y = 800
divide the 2nd by 5, and replace the trig values
√2/2 x - 3/2 y = 0
times 2
√2x - 3y = 0
√2x = 3y
sub that into the simplified first
3y + 3√3y = 800
y(3 + 3√3) = 800
y = 800/(3+3√3)
= 800/(3+3√3) * (3-3√3)/(3-3√3)
= 800(3)(1 - √3)/(9 - 27)
= (-400/3)(1 - √3) , after rationlizing
then
√2x = -400(1-√3)
times √2
2x = -400√2(1 - √3)
x = -200√2(1-√3)
approx. values:
x = 207.055
y = 97.607
Thank you very much, happy holidays
To solve these equations simultaneously, you can use trigonometric identities and the properties of sine and cosine to simplify and solve for x and y.
Let's start by expressing the trigonometric functions in terms of their known values. In this case, cosine 45 and sine 45 are both equal to √2/2, and cosine 30 and sine 30 are both equal to √3/2.
So, the original equations can be written as:
5x(√2/2) + 15y(√3/2) = 2,000 ---- (1)
5x(√2/2) - 15y(√3/2) = 0 ---- (2)
Now, let's multiply equation (2) by -1 to make the y coefficients have opposite signs:
-5x(√2/2) + 15y(√3/2) = 0 ---- (-2)
Now, add equations (1) and (-2):
5x(√2/2) + 15y(√3/2) - 5x(√2/2) + 15y(√3/2) = 2,000 + 0
Simplifying, we get:
30y(√3/2) = 2,000
Dividing both sides by 30(√3/2), we get:
y = 2,000 / (30(√3/2))
Simplifying further:
y = 2,000 / (15√3) = 40 / √3
Now substitute the value of y back into one of the original equations, let's use equation (1):
5x(√2/2) + 15(40/√3)(√3/2) = 2,000
Simplifying:
5x(√2/2) + 15(40/3) = 2,000
5x(√2/2) = 2,000 - 15(40/3)
5x(√2/2) = 2,000 - 200
Now divide by (√2/2):
5x = (2,000 - 200) / (√2/2)
x = [(2,000 - 200) / (√2/2)] / 5
Simplifying:
x = [(2,000 - 200) / (√2/2)] / 5 = 360 / (√2/2)
x = (360 * 2) / √2
x = 720 / √2 = 720 * (√2/2)
The value of (√2/2) = 1/√2, so we have:
x = 720 * (1/√2) = 720√2 / 2
x = 360√2
Therefore, the solution to the system of equations is:
x = 360√2
y = 40 / √3
Note: Sometimes, the values obtained can be long decimal numbers or irrational numbers due to the nature of the problem. In such cases, it is expected to have an approximation of the solution.