Solving special systems.

the system has infinitely many solutions

Step-by-step explanation:

Given the expressions

y - 2x = 4 1

2x -y - 4 = 0 2

We are to find the solution to the equation as shown

From equation 1: y = 4+2x

Substitute into 2:

2x - (4+2x) - 4 = 0

2x - 4 - 2x - 4 = 0

0 = 0

This shows that the equation has infinite number of solutions

Let y = k

Get x

Substitute y = k into equation 2

From 2: 2x -y - 4 = 0

2x - k - 4 = 0

2x = k+4

x = (k+4)/2 where k is any integer

Hence the system has infinitely many solutions

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step-by-step explanation:

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