The answer is 8 because
8 times 4 = 32
8 times 7 =56
8 times 3 =24
56 minus 24 = 32
The ordered pair (-1,5) is a solution to the system because it makes both equations true.
1) x+y=4
2) x-y=-6
1) -1 + 5 = 4
5-1 = 4 you can switch the order to make it easier to understand
so
4 = 4
2) -1 - 5 = -6
-1 + -5 = -6 you can change the sign in the middle and make y -
so
-6 = -6
Multiply total students by the percentage:
600 x 80% = 600 x 0.80 = 480
480 students want more field trips.
The tangent to
through (1, 1, 1) must be perpendicular to the normal vectors to the surfaces
and
at that point.
Let
. Then
is the level curve
. Recall that the gradient vector is perpendicular to level curves; we have

so that the gradient of
at (1, 1, 1) is

For the surface
, we have

so that
. We can obtain a vector normal to
by taking the cross product of the partial derivatives of
, and evaluating that product for
:


Now take the cross product of the two normal vectors to
and
:

The direction of vector (24, 8, -8) is the direction of the tangent line to
at (1, 1, 1). We can capture all points on the line containing this vector by scaling it by
. Then adding (1, 1, 1) shifts this line to the point of tangency on
. So the tangent line has equation
