The problem asks us to solve the system of equations: $\frac{2}{3}x - y = 2$ $3x - \frac{3}{4}y = 24$ and determine the nature of the solution and the relationship between the graphs of the equations.

1. Problem Description

The problem asks us to solve the system of equations:

$\frac{2}{3}x - y = 2$

$3x - \frac{3}{4}y = 24$

and determine the nature of the solution and the relationship between the graphs of the equations.

2. Solution Steps

First, we can multiply the first equation by 3 to eliminate the fraction:

$3(\frac{2}{3}x - y) = 3(2)$

$2x - 3y = 6$ (Equation 3)

Now, we can multiply the second equation by 4 to eliminate the fraction:

$4(3x - \frac{3}{4}y) = 4(24)$

$12x - 3y = 96$ (Equation 4)

Next, we can subtract Equation 3 from Equation 4 to eliminate $y$:

$(12x - 3y) - (2x - 3y) = 96 - 6$

$12x - 3y - 2x + 3y = 90$

$10x = 90$

$x = \frac{90}{10}$

$x = 9$

Now, substitute $x = 9$ into Equation 3:

$2(9) - 3y = 6$

$18 - 3y = 6$

$-3y = 6 - 18$

$-3y = -12$

$y = \frac{-12}{-3}$

$y = 4$

So the solution is $x = 9$ and $y = 4$, or the ordered pair $(9, 4)$. Since we found a unique solution, the equations are independent and consistent. Their graphs intersect at one point.

3. Final Answer

The solution to the system is (9, 4).

The equations are independent and consistent. Their graphs intersect at one point.