If f(x) is a linear function ,f(-1)=-3 , and f(4)=-5 , find an equation for f(x)

Step-by-step explanation:

The equation of the linear function: f(x) = mx + b

f(-1) = -3   ⇒   -3 = m(-1) + b  ⇒   m = b + 3

f(4)=-5   ⇒   -5 = m×4 + b  

                   -5 = 4(b+3) + b

                   -5 = 4b + 12 + b

                    5b = -5 - 12

                      b = -¹⁷/₅

m = -¹⁷/₅ + 3 = -¹⁷/₅ + ¹⁵/₅ = -²/₅

Therefore the function:

there are lots of ways.   pick's theorem tells us if the vertices are lattice points the area a is

a = b/2 + i - 1

where b is the number of boundary lattice points and i is the number of interior lattice points.

in the figure i count b=16, i=50 so

a = 8 + 50 - 1 = 57

let's try the shoelace formula.   the area is half the sum of the cross products of the sides.  

a(-2,6)   -2(2)-6(-5) = 26

b(-5,2)   -5(-3) - 2(-2) = 19

c(-2,-3) -2(-3) - (-3)(4) = 18

d(4,-3)   4(3) - (-3)(5) = 27

e(5,3)   5(3) - 3(1) = 12

f(1,3)     1(6) - 3(-2) = 12

a(-2,6)

each of the cross products is twice the area of the triangle with the associated side and one vertex the origin. so the area of abo=26/2=13.

area = (1/2) (26 + 19 +   18 + 27 + 12 + 12) = 57

that checks!

the ratio of balloons to tables is 5: 2 and the constant variation is 5.

the equation is y=5x