1:9 is the correct ratio. <3
Answer:
69
Step-by-step explanation:
Answer:
![C \cup B =[(x,y),:x=2 \:Or\: y=2 \:Or\: x+y=5, 1 \leq x \leq 6,1 \leq y \leq 6 ]\\](https://tex.z-dn.net/?f=C%20%5Ccup%20B%20%3D%5B%28x%2Cy%29%2C%3Ax%3D2%20%20%5C%3AOr%5C%3A%20%20y%3D2%20%5C%3AOr%5C%3A%20x%2By%3D5%2C%201%20%5Cleq%20x%20%5Cleq%20%206%2C1%20%5Cleq%20y%20%5Cleq%20%206%20%5D%5C%5C)
Step-by-step explanation:
Let the outcome from the two dice be (x,y) where x is the outcome from the red die and y is the outcome from the green die.
Event A: the red die shows 2
![A=[(x,y), x=2,1 \leq y \leq 6 ]](https://tex.z-dn.net/?f=A%3D%5B%28x%2Cy%29%2C%20x%3D2%2C1%20%5Cleq%20y%20%5Cleq%20%206%20%5D)
Event B: the numbers add to 5
![B=[(x,y),:x+y=5, 1 \leq x \leq 6,1 \leq y \leq 6 ]](https://tex.z-dn.net/?f=B%3D%5B%28x%2Cy%29%2C%3Ax%2By%3D5%2C%201%20%5Cleq%20x%20%5Cleq%20%206%2C1%20%5Cleq%20y%20%5Cleq%20%206%20%5D)
Event C: at least one of the numbers is 2;
![C=[(x,y),:x=2 \quad Or \quad y=2, 1 \leq x \leq 6,1 \leq y \leq 6 ]](https://tex.z-dn.net/?f=C%3D%5B%28x%2Cy%29%2C%3Ax%3D2%20%5Cquad%20Or%20%20%5Cquad%20y%3D2%2C%201%20%5Cleq%20x%20%5Cleq%20%206%2C1%20%5Cleq%20y%20%5Cleq%20%206%20%5D)
Event D: the numbers do not add to 9;
![D=[(x,y),:x+y\neq 9, 1 \leq x \leq 6,1 \leq y \leq 6 ]](https://tex.z-dn.net/?f=D%3D%5B%28x%2Cy%29%2C%3Ax%2By%5Cneq%209%2C%201%20%5Cleq%20x%20%5Cleq%20%206%2C1%20%5Cleq%20y%20%5Cleq%20%206%20%5D)
Therefore:
![C \cup B =[(x,y),:x=2 \:Or\: y=2 \:Or\: x+y=5, 1 \leq x \leq 6,1 \leq y \leq 6 ]\\](https://tex.z-dn.net/?f=C%20%5Ccup%20B%20%3D%5B%28x%2Cy%29%2C%3Ax%3D2%20%20%5C%3AOr%5C%3A%20%20y%3D2%20%5C%3AOr%5C%3A%20x%2By%3D5%2C%201%20%5Cleq%20x%20%5Cleq%20%206%2C1%20%5Cleq%20y%20%5Cleq%20%206%20%5D%5C%5C)
For this case we have that by definition, the standard form of a linear equation is given by:
![ax + by = c](https://tex.z-dn.net/?f=ax%20%2B%20by%20%3D%20c)
By definition, if two lines are perpendicular then the product of their slopes is -1. That is to say:
![m_ {1} * m_ {2} = - 1](https://tex.z-dn.net/?f=m_%20%7B1%7D%20%2A%20m_%20%7B2%7D%20%3D%20-%201)
We have the following point-slope equation of a line:
![y+4 = -\frac {2} {3}(x-12)](https://tex.z-dn.net/?f=y%2B4%20%3D%20-%5Cfrac%20%7B2%7D%20%7B3%7D%28x-12%29)
The slope is:
![m_ {1} = - \frac {2} {3}](https://tex.z-dn.net/?f=m_%20%7B1%7D%20%3D%20-%20%5Cfrac%20%7B2%7D%20%7B3%7D)
We find the slope
of a perpendicular line:
![m_ {2} = \frac {-1}{m_ {1}}\\m_ {2} = \frac {-1} {-\frac {2} {3}}\\m_ {2} = \frac{3} {2}](https://tex.z-dn.net/?f=m_%20%7B2%7D%20%3D%20%5Cfrac%20%7B-1%7D%7Bm_%20%7B1%7D%7D%5C%5Cm_%20%7B2%7D%20%3D%20%5Cfrac%20%7B-1%7D%20%7B-%5Cfrac%20%7B2%7D%20%7B3%7D%7D%5C%5Cm_%20%7B2%7D%20%3D%20%5Cfrac%7B3%7D%20%7B2%7D)
Thus, the equation is of the form:
![y-y_ {0} = \frac {3} {2} (x-x_ {0})](https://tex.z-dn.net/?f=y-y_%20%7B0%7D%20%3D%20%5Cfrac%20%7B3%7D%20%7B2%7D%20%28x-x_%20%7B0%7D%29)
We have the point through which the line passes:
![(x_ {0}, y_ {0}) :( 2, -3)](https://tex.z-dn.net/?f=%28x_%20%7B0%7D%2C%20y_%20%7B0%7D%29%20%3A%28%202%2C%20-3%29)
Thus, the equation is:
![y - (- 3) = \frac {3} {2} (x-2)\\y + 3 = \frac {3} {2} (x-2)](https://tex.z-dn.net/?f=y%20-%20%28-%203%29%20%3D%20%5Cfrac%20%7B3%7D%20%7B2%7D%20%28x-2%29%5C%5Cy%20%2B%203%20%3D%20%5Cfrac%20%7B3%7D%20%7B2%7D%20%28x-2%29)
We manipulate algebraically:
![y + 3 = \frac{3} {2} x- \frac {3} {2} (2)\\y + 3 = \frac{3} {2} x-3](https://tex.z-dn.net/?f=y%20%2B%203%20%3D%20%5Cfrac%7B3%7D%20%7B2%7D%20x-%20%5Cfrac%20%7B3%7D%20%7B2%7D%20%282%29%5C%5Cy%20%2B%203%20%3D%20%5Cfrac%7B3%7D%20%7B2%7D%20x-3)
We add 3 to both sides of the equation:
![y + 3 + 3 = \frac {3} {2} x\\y + 6 = \frac {3} {2} x](https://tex.z-dn.net/?f=y%20%2B%203%20%2B%203%20%3D%20%5Cfrac%20%7B3%7D%20%7B2%7D%20x%5C%5Cy%20%2B%206%20%3D%20%5Cfrac%20%7B3%7D%20%7B2%7D%20x)
We multiply by 2 on both sides of the equation:
![2(y + 6) = 3x\\2y + 12 = 3x](https://tex.z-dn.net/?f=2%28y%20%2B%206%29%20%3D%203x%5C%5C2y%20%2B%2012%20%3D%203x)
We subtract 3x on both sides:
![2y-3x + 12 = 0](https://tex.z-dn.net/?f=2y-3x%20%2B%2012%20%3D%200)
We subtract 12 from both sides:
![2y-3x = -12](https://tex.z-dn.net/?f=2y-3x%20%3D%20-12)
ANswer:
![-3x + 2y = -12](https://tex.z-dn.net/?f=-3x%20%2B%202y%20%3D%20-12)
2x+3=4x+2
We solve this equation:
2x+3=4x+2
2x-4x=2-3
-2x=-1
x=-1/-2
x=1/2
Therefore: we have only one solution.
Answer: This equation have one solution; x=1/2.