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.