Classify the system of equations and identify the number of solutions. need help

Question

Question 1: Classify the system of equations and identify the number of solutions.

35x + 5y = 15

y = 3 − 7x

A.) consistent, independent; one

B.) consistent, dependent; one

C.) inconsistent; none

D.) consistent, dependent; infinite

Question 2: Minimize the objective function P = 45x + 48y for the given constraints.

x ≥ 0

y ≥ 0

5x + y ≥ 16

x + 3y ≥ 6

A.) 768

B.) 84

C.) 183

D.) 270

hello can please explain and solve.

Answer 1

For Question 1:

35x + 5y = 15 [subtract 35x from both sides to get:

5y = 15 - 35x [divide both sides by 5 to get:

y = 3 - 7x

Notice that this is the same as the second equation so these are the same line. That means they are consistent but that they are dependent with an infinite number of common solutions.

For Question 2:

5x + y ≥ 16 is the same as y ≥ -5x + 16

And x + 3y ≥ 6 is the same as y ≥ -1/3x + 2

The x ≥ 0 and y ≥ 0 tells us we are above the x-axis and to the right of the y-axis. Graphing this gives:

For the function P = 45x + 48y, obviously there is no maximum value of P because x and y can be as big as we want them to ne so the value of P can grow without bound. The SMALLEST value of P would be limited by the constraints y ≥ -5x + 16 and y ≥ -1/3x + 2 graphed above. We can see that the critical points are those I've listed on the graph as the "corner points" which are (0, 16), (3, 1), and (6, 0). We just need to try out those points in the function P = 45x + 48y to see which one gives the smallest value of P.

P(0,16) = 45(0) + 48(16) = 768

P(3,1) = 45(3) + 48(1) = 183

P(6,0) = 45(6) + 48(0) = 270.

So the smallest value of P is 183 and it occurs when x = 3 and y = 1

Example: The point (2,6) is slightly above and to the left of (3,1) and it gives a value of P = 45(2) + 48(6) = 378. The point (5, 1/3) is a point slightly to the right and down from (3,1) and it gives a value of P = 45(5) + 48(1/3) = 241