1)Find the exact solutions of the system of equations x^2 + 2y^2 = 18 and x = 2y.
x=2y
x^2+2y^2=18
(2y)^2+2y^2=18
4y^2+2y^2=18
6y^2=18
y^2=3
y=+-�ã3
x=2(+-�ã3)
x=+-2�ã3
(+-2sqrt3,+-sqrt3)
2)Simplify, (d/(d^2-9)) + (5/(2d+6))
[d / (d^2 - 9)] + [5 / (2d + 6)]
= [d / (d - 3) (d +3)] + [5 / 2 (d + 3)]
= [2d + (5 {d - 3})] / [2 (d + 3) (d - 3)]
= [2d + 5d - 15] / [2 (d + 3) (d - 3)]
= (7d - 15) / [2 (d + 3) (d - 3)]
= (7d - 15) / (2d^2 - 18)
To find the exact solutions of the system of equations x^2 + 2y^2 = 18 and x = 2y, we can solve the equations simultaneously.
1. Start with the equation x = 2y.
2. Substitute the value of x in the other equation:
(2y)^2 + 2y^2 = 18.
3. Simplify the equation:
4y^2 + 2y^2 = 18.
6y^2 = 18.
4. Divide both sides of the equation by 6:
y^2 = 3.
5. Take the square root of both sides:
y = ±√3.
6. Substitute the value of y in the equation x = 2y:
x = 2(±√3).
x = ±2√3.
Therefore, the exact solutions of the system of equations are (±2√3, ±√3).
Now, let's simplify the expression (d/(d^2-9)) + (5/(2d+6).
1. Factor the denominator d^2 - 9:
(d - 3)(d + 3).
2. Rewrite the expression with the factored form:
d / (d - 3)(d + 3) + 5 / (2d + 6).
3. Simplify the denominators:
d / (d - 3)(d + 3) + 5 / 2(d + 3).
4. Find the common denominator: (d - 3)(d + 3)(2).
5. Multiply the numerators by the appropriate factors:
(d * 2) + (5 * (d - 3)).
(2d + 5d - 15).
7d - 15.
6. Write the result over the common denominator:
(7d - 15) / (2(d - 3)(d + 3)).
(7d - 15) / (2d^2 - 18).
Therefore, the simplified expression is (7d - 15) / (2d^2 - 18).