Determine the ordered pair that solves the linear system

3(x – 1) – 2(y – 3) = 0 and 3(x + 4) – 4(y – 1) – 1 = 0.

(3,6)

To determine the ordered pair that solves the linear system, we need to find the values of x and y that satisfy both equations simultaneously. Here's how you can solve the system of equations step by step:

Step 1: Simplify the equations
Start by simplifying both equations by distributing and combining like terms:

1) 3(x – 1) – 2(y – 3) = 0
3x - 3 - 2y + 6 = 0
3x - 2y + 3 = 0 (Equation A)

2) 3(x + 4) – 4(y – 1) – 1 = 0
3x + 12 - 4y + 4 - 1 = 0
3x - 4y + 15 = 0 (Equation B)

Step 2: Solve for one variable
Choose one equation and solve it for one variable in terms of the other variable. Let's solve Equation A for x:

3x - 2y + 3 = 0
3x = 2y - 3
x = (2y - 3) / 3 (Equation C)

Step 3: Substitute the variable
Substitute the expression for x in terms of y from Equation C into the other equation (Equation B):

3x - 4y + 15 = 0
3[(2y - 3) / 3] - 4y + 15 = 0
2y - 3 - 4y + 15 = 0
-2y + 12 = 0
2y = 12
y = 6

Step 4: Solve for the other variable
Now that we have the value of y, substitute it back into Equation C to solve for x:

x = (2y - 3) / 3
x = (2(6) - 3) / 3
x = (12 - 3) / 3
x = 9 / 3
x = 3

Step 5: Write the ordered pair
The ordered pair that solves the linear system is (x, y) = (3, 6).

Therefore, the ordered pair (3, 6) is the solution to the given linear system of equations.