The system of equations in the image is:

$4x + 3y = 5$ $2x - \frac{3}{4}y = 1$

Let's solve this system using the substitution or elimination method. We will proceed with the elimination method by eliminating one variable and solving for the other.

Step 1: Eliminate $x$

First, we need to make the coefficients of $x$ the same in both equations. We can multiply the second equation by 2 to match the coefficient of $x$ in the first equation.

Multiply the second equation by 2:

$2(2x - \frac{3}{4}y) = 2(1)$ $4x - \frac{3}{2}y = 2$

Now we have the system:

4x + 3y = 5 \tag{1} 4x - \frac{3}{2}y = 2 \tag{2}

Step 2: Subtract equation (2) from equation (1)

Subtract equation (2) from equation (1) to eliminate $x$:

$(4x + 3y) - (4x - \frac{3}{2}y) = 5 - 2$

Simplify the equation:

$4x + 3y - 4x + \frac{3}{2}y = 3$

$3y + \frac{3}{2}y = 3$

Combine the terms on the left:

$\frac{6}{2}y + \frac{3}{2}y = 3$

$\frac{9}{2}y = 3$

Step 3: Solve for $y$

Multiply both sides by 2 to get rid of the fraction:

$9y = 6$

Solve for $y$:

$y = \frac{6}{9} = \frac{2}{3}$

Step 4: Substitute $y = \frac{2}{3}$ back into one of the original equations

Substitute $y = \frac{2}{3}$ into equation (1):

$4x + 3\left(\frac{2}{3}\right) = 5$

Simplify:

$4x + 2 = 5$

Solve for $x$:

$4x = 5 - 2$

$4x = 3$

$x = \frac{3}{4}$

Final Solution:

The solution to the system is:

$x = \frac{3}{4}, \quad y = \frac{2}{3}$

Would you like further clarification or details on any of these steps?

Here are some related questions for deeper understanding:

1. How can you solve this system using substitution instead of elimination?
2. What happens if the system of equations has no solution or infinitely many solutions?
3. How do you interpret the graphical representation of this system of equations?
4. How would the solution change if the coefficients in the system were different?
5. Can you generalize the elimination method for systems with more variables?

Tip: Always verify the solution by substituting both $x$ and $y$ into the original equations to ensure accuracy.