The image contains a system of linear equations to solve for $x$ and $y$. The equations are:

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

We are asked to solve for $x = a$ and $y = b$, and then determine the value of $a - 3b$.

Let me solve this system of equations step by step.

1. Eliminate the fractions

To make the calculations easier, we can multiply both equations by the least common denominator of the fractions involved.

• The first equation: Multiply by 12 (LCM of 4 and 3) $12 \times \left( \frac{3}{4}x - \frac{1}{3}y \right) = 12 \times 5$ This simplifies to: $9x - 4y = 60$

• The second equation: Multiply by 6 (LCM of 2 and 3) $6 \times \left( \frac{1}{2}x + \frac{2}{3}y \right) = 6 \times (-2)$ This simplifies to: $3x + 4y = -12$

2. Add the equations to eliminate $y$

Now, we can add both equations: $(9x - 4y) + (3x + 4y) = 60 + (-12)$ This simplifies to: $12x = 48$ So, we find: $x = 4$

3. Substitute $x = 4$ into one of the equations

Let's substitute $x = 4$ into the second equation: $\frac{1}{2}x + \frac{2}{3}y = -2$ Substitute $x = 4$: $\frac{1}{2}(4) + \frac{2}{3}y = -2$ This simplifies to: $2 + \frac{2}{3}y = -2$ Subtract 2 from both sides: $\frac{2}{3}y = -4$ Multiply both sides by $\frac{3}{2}$: $y = -6$

4. Calculate $a - 3b$

We now have $a = 4$ and $b = -6$. We need to find $a - 3b$: $a - 3b = 4 - 3(-6) = 4 + 18 = 22$

Final Answer

The value of $a - 3b$ is $22$.

Would you like further details or have any other questions? Here are 5 related problems you might want to try:

1. Solve a similar system of linear equations using substitution method.
2. Solve the system $2x - 3y = 7$ and $4x + y = 1$.
3. What happens if both equations are multiplied by different constants and then subtracted?
4. Can you solve a system of three variables like $x + y + z = 6$, $2x - y + z = 3$, and $x - 2y + 2z = -1$?
5. What are the conditions for a system of linear equations to have no solution or infinite solutions?

Tip: When solving systems of linear equations, multiplying to eliminate fractions can simplify calculations and reduce the chance of errors.