Solving equations simultaneously:
.7*x+.30*y=33.1
(x*y)/(.7*y+.30*x) = 3.66
I get stuck after I get 51.972=45.2y-.42y^2
I know the answers are supposed to be
x=2.6 and y=104
To solve the given system of equations simultaneously, you can use the substitution method. Here are the steps:
1. Start with the first equation:
0.7x + 0.3y = 33.1 ----(equation 1)
2. Solve equation 1 for x:
0.7x = 33.1 - 0.3y
x = (33.1 - 0.3y) / 0.7
3. Substitute the value of x in the second equation:
[(33.1 - 0.3y) / 0.7 * y] / (0.7y + 0.3[(33.1 - 0.3y) / 0.7]) = 3.66
4. Simplify the equation:
[(33.1 - 0.3y) * y] / (0.7y + (0.3(33.1 - 0.3y)) / 0.7) = 3.66
Multiply both sides by 0.7 to eliminate the denominator:
[0.7(33.1 - 0.3y) * y] = 3.66(0.7y + 0.3(33.1 - 0.3y))
Simplify further:
(0.7y - 0.21y^2) = 2.562y + 2.156 - 0.06798y^2
5. Rearrange the equation in descending order of powers of y:
-0.06798y^2 - 0.21y^2 + 0.7y - 2.562y - 2.156 = 0
6. Combine similar terms and move all the terms to one side:
-0.27898y^2 - 1.862y - 2.156 = 0 ----(equation 2)
7. This is a quadratic equation. To solve it, you can use the quadratic formula, which states that for an equation of the form ax^2 + bx + c = 0, the solutions for x are given by:
x = (-b ± √(b^2 - 4ac)) / (2a)
In our case, the equation is -0.27898y^2 - 1.862y - 2.156 = 0. Comparing with the quadratic formula, we can find the values of a, b, and c:
a = -0.27898
b = -1.862
c = -2.156
8. Substitute the values of a, b, and c into the quadratic formula and solve for y:
y = [-(-1.862) ± √((-1.862)^2 - 4(-0.27898)(-2.156))] / (2(-0.27898))
Simplify further:
y = (1.862 ± √(3.465344 + 0.242474)) / (-0.55796)
y = (1.862 ± √3.707818) / (-0.55796)
9. Calculate the values inside the square root and perform the addition/subtraction:
y = (1.862 ± 1.925) / (-0.55796)
This gives two possible values for y:
i) y = (1.862 + 1.925) / (-0.55796) ≈ -5.448
ii) y = (1.862 - 1.925) / (-0.55796) ≈ 104.15
10. Substitute each value of y back into equation 1 to find the corresponding values of x:
For y = -5.448:
x = (33.1 - 0.3(-5.448)) / 0.7 ≈ 9.162
For y = 104.15:
x = (33.1 - 0.3(104.15)) / 0.7 ≈ 2.6
Therefore, the solutions to the system of equations are x = 2.6 and y = 104 (rounded to two decimal places).