A linear function that represents the number of animals adopted from the shelter is compared to a different linear function that represents the hours volunteers work at the shelter each week. Describe the key features of the functions that are needed to determine if these lines intersect.

The slopes of the equations must be different:

y1= a*x + b

y2 = c*x + b

then a ≠ c

Step-by-step explanation:

A linear function is:

Y = a*x + b

now, suppose we have other linear function

H = c*x + d

Let's find the condition needed to have an intersection of these two lines, or:

a*x + b = c*x + d

(a - c)*x = d - b

Where a, c, d and b are real numbers.

You can see that the only condition is that the coefficient that multiplicates the variable (x) must be different than 0 (because if it is 0, the x vanishes and we can never solve the equation)

So we must have that the slopes of the two linear equations need to be different.