The following system has one solution: x = 2, y = −2, and z = 3. 4x − 2y + 5z = 27 equation 1 x + y = 0 equation 2 −x − 3y + 2z = 10 equation 3 (a) solve the system provided by equations 1 and 2. (if there is no solution, enter no solution. if the system has an infinite number of solutions, express x, y, and z in terms of the parameter t.)

x = t

y = -t

z = (17-6t)/5

Step-by-step explanation:

First we will write down Equation 1 and Equation 2

4x-2y+5z = 17(1)

x + y = 0(2)

Since we are asked to express our answer in terms of the parameter t so lets suppose x = t and  y = -t

Now substitute the value of x and y in equation (1) we get

4(t) -2(-t) +5z = 17

6t + 5z = 17

5z = 17 -6t

z = (17-6t)/5

hence we get the following answer

x = t

y = -t

z= (17-6t)/5

we can also verify our answer by substituting the values of x ,y and z in any of the two equations above, for example lets substitute these values in equation 1

4(t) -2(-t) + 5((17-6t)/5)) = 17

6t + 17 -6t = 17

6t -6t = 17 -17

0 = 0

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

17 1/3

step-by-step explanation: