The system is:
i) <span>2x – 3y – 2z = 4
ii) </span><span>x + 3y + 2z = –7
</span>iii) <span>–4x – 4y – 2z = 10
the last equation can be simplified, by dividing by -2,
thus we have:
</span>i) 2x – 3y – 2z = 4
ii) x + 3y + 2z = –7
iii) 2x +2y +z = -5
The procedure to solve the system is as follows:
first use any pairs of 2 equations (for example i and ii, i and iii) and equalize them by using one of the variables:
i) 2x – 3y – 2z = 4
iii) 2x +2y +z = -5
2x can be written as 3y+2z+4 from the first equation, and -2y-z-5 from the third equation.
Equalize:
3y+2z+4=-2y-z-5, group common terms:
5y+3z=-9
similarly, using i and ii, eliminate x:
i) 2x – 3y – 2z = 4
ii) x + 3y + 2z = –7
multiply the second equation by 2:
i) 2x – 3y – 2z = 4
ii) 2x + 6y + 4z = –14
thus 2x=3y+2z+4 from i and 2x=-6y-4z-14 from ii:
3y+2z+4=-6y-4z-14
9y+6z=-18
So we get 2 equations with variables y and z:
a) 5y+3z=-9
b) 9y+6z=-18
now the aim of the method is clear: We eliminate one of the variables, creating a system of 2 linear equations with 2 variables, which we can solve by any of the standard methods.
Let's use elimination method, multiply the equation a by -2:
a) -10y-6z=18
b) 9y+6z=-18
------------------------ add the equations:
-10y+9y-6z+6z=18-18
-y=0
y=0,
thus :
9y+6z=-18
0+6z=-18
z=-3
Finally to find x, use any of the equations i, ii or iii:
<span>2x – 3y – 2z = 4
</span>
<span>2x – 3*0 – 2(-3) = 4
2x+6=4
2x=-2
x=-1
Solution: (x, y, z) = (-1, 0, -3 )
Remark: it is always a good attitude to check the answer, because often calculations mistakes can be made:
check by substituting x=-1, y=0, z=-3 in each of the 3 equations and see that for these numbers the equalities hold.</span>
Answer:
26 rows
Step-by-step explanation:
this is like a rectangle length×width situation.
seats per row = s
number of rows = r
s × r = 884
s = r + 8
so, we can use e.g. the second equation in the first :
(r + 8) × r = 884
r² + 8r = 884
r² + 8r - 884 = 0
the general solution to such a quadratic equation is
x = (-b ± sqrt(b² - 4ac))/(2a)
in our case
x = r
a = 1
b = 8
c = -884
r = (-8 ± sqrt(8² - 4×1×-884))/(2×1) =
= (-8 ± sqrt(64 + 3536))/2 = (-8 ± sqrt(3600))/2 =
= (-8 ± 60)/2 = -4 ± 30
r1 = -4 + 30 = 26
r2 = -4 - 30 = -34
a negative number did not make any sense for the number of rows, so r = 26 is our answer.
Answer: 27434
Step-by-step explanation:
Given : Total number of vials = 56
Number of vials that do not have hairline cracks = 13
Then, Number of vials that have hairline cracks =56-13=43
Since , order of selection is not mattering here , so we combinations to find the number of ways.
The number of combinations of m thing r things at a time is given by :-
Now, the number of ways to select at least one out of 3 vials have a hairline crack will be :-
Hence, the required number of ways =27434
(-4,-2)
You just plug in the numbers from the ordered pair, into your equation. Remember, the formula for ordered pairs are (x,y). :)
If you would like to solve the equation k / 2 + 9 = 30, you can calculate this using the following steps:
k / 2 + 9 = 30
k / 2 = 30 - 9
k / 2 = 21 /*2
k = 21 * 2
k = 42
The result is 42.
