Find the slope of the line passing through the points at (a, a+3) and (4a, a-5).

2) Together, two printers can print 7500 lines if the first printer prints for 2 minutes and the second prints for 1 minute. If the first printer prints for 1 minute and the second prints for 2 minutes, they can print 9000 lines together. Find the number of lines per minute that each printer prints.

To find the slope of a line passing through two points, you can use the formula:

slope = (y2 - y1) / (x2 - x1)

1) For the first problem, we are given the coordinates of two points: (a, a+3) and (4a, a-5).

Let's substitute these values into the formula:

slope = (a-5 - (a+3)) / (4a - a)

Simplifying the numerator gives:

slope = (a - 5 - a - 3) / (4a - a)
= (-8) / (3a)

So the slope of the line passing through the points (a, a+3) and (4a, a-5) is -8 / (3a).

2) For the second problem, let's denote the number of lines per minute that the first printer prints as x and the number of lines per minute that the second printer prints as y.

From the given information, we can create two equations:

Equation 1: 2x + y = 7500
Equation 2: x + 2y = 9000

To solve this system of equations, we can use any method such as substitution or elimination. Let's use the substitution method.

Rearrange Equation 1 to solve for y:

y = 7500 - 2x

Substitute this value of y into Equation 2:

x + 2(7500 - 2x) = 9000

Simplify the equation:

x + 15000 - 4x = 9000
-3x = -6000
x = 2000

Now substitute this value of x back into Equation 1 to find y:

2(2000) + y = 7500
4000 + y = 7500
y = 3500

Therefore, the first printer prints 2000 lines per minute, and the second printer prints 3500 lines per minute.