Solving Three Simultaneous Equations Help Please?

Question

Hi. could someone please show me the full working out for the following 4 questions... i have heaps of dificulties with them.. i have done 18so far but i need help with these last 5.. thanks heaps! i already have the answers... but working out i need to see how its done. thanks

5v) 2x+y+z=10
x+5y-3z=-7
x+2y+3z=23

5u) 5x + z = 2
y + z =5
-x + y = 1

5x) x- 2y + 2z = 8
3x - 2y + z = 8
2x + y - z = 1

5y) x+3y+2z = 1
2x - y - z = 6
3x + y - 4z=-1

6) A parabola with the queation y = ax^2 + bx + c, with the points (-1,-10), (2,-1) and, (3,-6), find; a,b,c

Answer 1

to solve a system of 3 equations to three unknowns you have to replace one of the unknowns by his relation to the other 2 unknowns...I will solve the first system explaning every step, you will have to do the others.

2x + y + z = 10 => z = 10 - 2x - y------(1)
x + 5y - 3z = -7---------(2)
x + 2y + 3z = 23--------(3)

replacing z by its value (1) in equations (2) and (3) we get
(2) => x + 5y - 3(10 - 2x - y) = -7
=> x + 5y - 30 + 6x + 3y = -7
=> 7x + 8y = 23

(3) => x + 2y + 3(10 - 2x - y) = 23
=> x + 2y + 30 - 6x - 3y = 23
=> -5x - y = -7 => 5x + y = 7
we have a system of two equations, 2 unknowns:
7x + 8y = 23
5x + y = 7
solving by elimination
7x + 8y = 23 ----(a)
-8* 5x + (-8) * y = -8*7 => -30x - 8y = -56------(b)
(a) + (b) =>
7x - 30x + 8y - 8y = 23 - 56
-23x = -33
x=33/23
replacing x in (b) we get : y = 7 - 5x = 7 - 165/23 = -4/23
and replacing in (1) we get :
z = 10 -2x - y = 10 - 2(33/23) - (-4/23)= 168/23

solve the other systems the same way, by replacing z from the first equation in the 2nd and 3rd equation, then you will get a system with x and y that you can solve and get all three unknowns.

in the question (6) I will give you the systems and you can solve them
each point on the parabola (x,y) is to be replaced in the equation
y = ax^2 + bx + c
-10 = a + b + c
-1 = 4a + 2b + c
-6 = 9a + 3b + c

Good Luck

Answer 2

5v)
2x + y + z = 10
x + 5y - 3z = - 7
x + 2y + 3z = 23
6x + 3y + 3z = 30
7x + 8y = 23
2x + 7y = 16
- 14x - 16y = - 46
14x + 49y = 112
33y = 66
y = 2
2x + 14 = 16
x = 1
1 + 4 + 3z = 23
z = 6
2 + 2 + 6 = 10

5u)
5x + z = 2
y + z =5
-x + y = 1
5x - y = -3
4x = -2
x = -1/2
y = 1/2
z = 9/2
-5/2 + 9/2 = 2

5x)
x - 2y + 2z = 8
3x - 2y + z = 8
2x + y - z = 1
4x + 2y - 2z = 2
5x - y = 7
5x = 10
x = 2
y = 3
z = 8
6 - 6 + 8 = 8

5y)
x + 3y + 2z = 1
2x - y - z = 6
3x + y - 4z = - 1
6x - 3y - 3z = 18
7x - z = 19
5x - 5z = 5
-x + z = - 1
6x = 18
x = 3
z = 2
6 - y - 2 = 6
y = -2
3 - 6 + 4 = 1

a - b + c = - 10
4a + 2b + c = -1
9a + 3b + c = -6
5a + b = -5
3a + 3b = 9
-a - b = -3
4a = -8
a = -2
b = 5
c = -10 + 5 + 2
c = -3
- 8 + 10 - 3 = -1

Answer 3

3a - 5b - 7c =0 ~~(a million eqauation) -4a - b + 6c=0 and we may be able to write this 4a +b - 6c =0 ~~(2) (2)*5 20b + 5b -30c =0~~(3) (a million) + (3) 3a - 5b -7c +20a +5b -30c =0 23a =37c c= 23a/37 u can fiend yet another uncoun fee... :-)