Find the three unknowns...?

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

do your own homework. otherwise you won't learn a thing.
I'll do the first problem only, and will show you the steps I took.
LEARN from it.

problem: 5v
2x + y + z = 10 ............ (I)
x + 5y - 3z = -7 ............ (II)
x + 2y + 3z = 23 ............ (III)

start with (II) & (III)... eliminate the "x" variable by subtracting the two:
x + 5y - 3z = -7
-x - 2y - 3z = -23
------------------------
0 + 3y - 6z = -30

divide this eq by 3 to simplify:

y - 2z = -10 .......... (IV)

use (I) and (II) to eliminate another "x" variable from an independent set:

2x + y + z = 10
-2x - 10y + 6z = 14 multiply (II) by -2
--------------------------
0 - 9y + 7z = 24

-9y + 7z = 24 ......... (V)

Now that we have eliminated "x" on two independent sets, we can solve for y and z using eq. (IV) and (V): get rid of the "y" variable.

9y - 18z = -90 multiply (IV) by 9
-9y + 7z = 24
-------------------
0 - 11z = -66

z = 6 .......... answer for z

now lets hunt for y by substituting z into (IV):

y - 2(6) = -10
y = -10 + 12
y = 2 ......... answer for y

now using the values of y=2 and z=6... lets go for the x-variable using eq (I):

2x + 2 + 6 = 10
2x = 10 - 8 = 2
x = 1 .......... answer for x

to sum it up:
x = 1
y = 2
z = 3

the steps were long, but it's good to learn the basic steps and to realize that you need N independent equations to solve for N variables.

Answer 2

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

Let's form a two equation two unknown by taking the third equation, solving for x, and then plugging this into the first and second equations.

x = -2y - 3z + 23.
Plugging this into the first equation gives us:

2(-2y - 3z + 23) + y + z = 10
-4y - 6z + 46 + y + z = 10
-3y - 5z = -36

Plugging this into the second equation gives us

(-2y - 3z + 23) + 5y - 3z = -7
3y - 6z = -30
y - 2z = -10

So our two equations two unknowns system is:

-3y - 5z = -36
y - 2z = -10

And now we can solve this however we like. I'm going to use substitution. y = 2z - 10, so

-3(2z - 10) - 5z = -36
-6z + 30 - 5z = -36
-11z = -66
z = 6

Now that we have the value of z, we can get y.
y = 2z - 10 = 2(6) - 10 = 12 - 10 = 2.

So y = 2, z = 6. We can easily get x now.
x = -2y - 3z + 23
x = -2(2) - 3(6) + 23
x = -4 - 18 + 23
x = 1

So x = 1, y = 2, z = 6.

Now do the rest on your own using the same process.