What is the shortest distance from the surface xy+3x+z2=9xy+3x+z2=9 to the origin?

Shortest distance is (2,1,1)

Step-by-step explanation:

Using the formula for distance. D² = x² + y² + z²

Minimizing D² is just as valid as minimizing D.

Now, let's rearrange the original equation to get z² = 9 - xy - 3x

Let's put this into the equation for D² to obtain;

D² = x² + y² + 9 - xy - 3x

Taking the partial derivatives with respect to x and y and set them equal to 0 to get the minimum, we have;

dD²/dx = 2x - y - 3 = 0

Also;

dD²/dy = 2y - x = 0

Solve dD²/dy for x, we have:

x = 2y

Put this into the first equation to obtain: 2(2y) - y = 3

This gives ; 3y = 3

So y = 1 and x = 2

Let's plug in y = 1 and x = 2 into

z² = 9 - xy - 3x

Thus gives;

z² = 9 - (1 x 2) - (3 x 2)

z² = 9 - 2 - 6

z² = 1

And z = 1

So the closest point is (2, 1, 1)

step-by-step explanation:

we are given a graph for system of inequalities.

in order to write the system of inequalities for the given graph, we need to find the inequality for each of the line separately.

let us find the slope of dotted line first.

taking two coordinates on the graph.

let us say (0,0) and (3,1).

let us find the slope of the line now.

slope of the dotted line would be =.

y-intercept of dotted line is y=0 only.

plugging slope m= and b=0 in slope-intercept form y=mx+b, we get

.

the line is shaded up of that dotted line.

therefore, for shading up we would take greater than inequality symbol > .

and because it's a dotted line we would take only > symbol in place of = sign in above equation of line.

therefore, inequality for dotted line would be

let us take two points (3,1) and (0,4) on the solid line.

slope of the solid line would be =.

for the solid line we have y-intercept b= 4.

plugging slope m= -1 and b=4 in slope-intercept form y=mx+b, we get

y = -1x + 4.

the line is shaded down of that solid line.

therefore, for shading down we would take less than inequality symbol < .

and because it's a solid line we would take ≤ symbol in place of = sign in above equation of line.

therefore, inequality for solid line would be