Three times the 1st number plus the 2nd number plus twice the 3rd is 5. if three times the 2nd number is subtracted from the sum of the 1st and three times the 3rd number, the result is 2. if the 3rd number is subtracted from two times the 1st number and three times the 2nd, giving a result of 1. find the three numbers.

The first number is 5, the second number is -4 and the third number is -3.

Let x represent the first number, y represent the second number and z represent the third number.

Three times the 1st number plus the 2nd number plus twice the 3rd is 5. Hence:

3x + y + 2z = 5    (1)

Three times the 2nd number is subtracted from the sum of the 1st and three times the 3rd number, the result is 2. Hence:

x + 3z - 3y = 2     (2)

The 3rd number is subtracted from two times the 1st number and three times the 2nd, giving a result of 1. Hence:

2x + 3y - z = 1    (3)

Solving equations 1, 2 and 3 simultaneously gives:

x = 5, y = -4, z = -3.

The first number is 5, the second number is -4 and the third number is -3.

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c = 3,

b = 2

a = -1

Step-by-step explanation:

I want to start by saying let me know if there is anything you don't get.  I'd be happy to explain.

The best way to solve this, as with most math problems, is to write things out, or draw a picture if possible.  So let's call the first number a, second number band third c.  

Now let's write out what is described.

3a + b + 2c = 5

a + 3c - 3b = 2

2a + 3b - c = 1

So, since you are given three variables and three equations it indicates you should be able to find a pretty nice answer.  It's possible  things may not work out still, so keep that in mind.  Anyway, this is the method I liek to use with most systems of equations.

Taking the first equation, 3a + b + 2c = 5, we want to solve for one of the variables.  Since b doesn't have a coefficient let's do that.  So we want to subtract 3a and 2c from both sides to get b by itself.

3a + b + 2c = 5

b + 2c = 5 - 3a

b = 5 - 3a - 2c

Now we move on to the second equation.  a + 3c - 3b = 2, First we want to plug what we got or b in the last part in for b here.  so we got b = 5 - 3a - 2c so now a + 3c - 3b = 2 becomes a + 3c - 3(5 - 3a - 2c) = 2.  Let's simplify this real quick.

a + 3c - 3(5 - 3a - 2c) = 2

a + 3c - 15 + 9a + 6c = 2

-15 + 10a + 9c = 2

Now from here we want to get one of the variables by itself again.  This time let's do a.

-15 + 10a + 9c = 2

10a = 17 - 9c Now divide both sides by 10

a = 1.7 - .9c

We can also get our b = equation in terms of c as well, so I'll do that.

b = 5 - 3a - 2c

b = 5 - 3(1.7 - .9c) - 2c

b = 5 - 5.1 + 2.7c - 2c

b = -.1 + .7c

Now we tackle the third equation.  We want to plug in the b= term and a= term and then get c to just equal a number.  It might be confusing to explain so hopefully seeing the process will help.

2a + 3b - c = 1 Now use b = -.1 + .7c and a = 1.7 - .9c

2(1.7 - .9c) + 3(-.1 + .7c) - c = 1

3.4 - 1.8c - .3 + 2.1c - c = 1

3.1 - .7c = 1

-.7c = -2.1

c = 3

Now we work backwards and plug c into the b= term and the a= term.

b = -.1 + .7c now plug in c = 3

b = -.1 + .7(3)

b = -.1 + 2.1

b = 2

a = 1.7 - .9c Again plug in 3 for c

a = 1.7 - .9(3)

a = 1.7 - 2.7

a = -1

So c = 3, b = 2 and a = -1 should be the answer.  make sure to plug each of these into the starting three equations to check.  let me know if you didn't understand any of this, it was a lng explanation.  There is also a matrix method you could use, if you are familiar with matrices, and I'd be happy to explain that as well.  

the value of v.w= -12.

step-by-step explanation:

we are given three vector r,v and w as:

r = < 5, -5, -2> ; v = < 2, -8, -8> ; w = < -2, 6, -5>

we know that r,v and w could also be represented as:

r=5i-5j-2k

v=2i-8j-8k

and w= -2i+6j-5k

where i,j and k are the unit vectors of x,y and z-axis respectively.

also we know that it has the following property:

i.i=1 ; j.j=1; k.k=1

and i.=j.i=i.k=k.i=j.k=k.j=0

i.e. when we multiply two vectors or we find it's dot product we need to only multiply the components of x coordinates with each other; y with each other and z with each other.

hence, v.w= (2i-8j- -2i+6j-5k)

                  = 2×(-2)+(-8)×(6)+(-8)×(-5)

                  = -4-48+40

                  = -12

hence, v.w= -12