Let’s solve the given system of equations step by step.

Step 1: Write the equations in a simplified form.

1. From the first equation:

$\frac{x + 8}{4} + \frac{y - 1}{3} = 1$

Multiply through by the least common multiple (LCM) of the denominators, which is 12:

$12 \cdot \frac{x + 8}{4} + 12 \cdot \frac{y - 1}{3} = 12 \cdot 1$

$3(x + 8) + 4(y - 1) = 12$

Simplify:

$3x + 24 + 4y - 4 = 12$

$3x + 4y + 20 = 12$

$3x + 4y = -8$

So, the first equation simplifies to:

$3x + 4y = -8 \quad \text{(Equation 1)}.$

2. The second equation is already in standard form:

$2x - y = 24 \quad \text{(Equation 2)}.$

Step 2: Eliminate one variable.

We have the system:

$3x + 4y = -8 \quad \text{(1)}$ $2x - y = 24 \quad \text{(2)}.$

To eliminate $y$, let’s multiply Equation (2) by 4 so the $y$-terms have the same coefficient:

$4(2x - y) = 4(24)$

$8x - 4y = 96 \quad \text{(Modified Equation 2)}.$

Now, add Equation (1) and the modified Equation (2):

$(3x + 4y) + (8x - 4y) = -8 + 96$

$11x = 88$

Solve for $x$:

$x = \frac{88}{11} = 8$

Step 3: Solve for $y$.

Substitute $x = 8$ into Equation (2):

$2(8) - y = 24$

$16 - y = 24$

$y = 16 - 24 = -8$

Step 4: Verify the solution.

1. Substitute $x = 8$ and $y = -8$ into the first equation:

$\frac{x + 8}{4} + \frac{y - 1}{3} = 1$

$\frac{8 + 8}{4} + \frac{-8 - 1}{3} = 1$

$\frac{16}{4} + \frac{-9}{3} = 1$

$4 - 3 = 1$

This is correct.

2. Substitute $x = 8$ and $y = -8$ into the second equation:

$2x - y = 24$

$2(8) - (-8) = 24$

$16 + 8 = 24$

This is also correct.

Final Answer:

$(x, y) = (8, -8)$

Would you like further explanation or a step-by-step breakdown of any specific part?

5 Related Questions:

1. How can the substitution method be used to solve this system instead of elimination?
2. What if the system had no solution? How would that be determined?
3. What if the system had infinitely many solutions? How would they be expressed?
4. Can this problem be solved graphically? What would the solution look like on a graph?
5. How does the presence of fractions in the original system affect the solving process?

Tip:

When working with fractions in equations, always eliminate them first by multiplying through by the LCM of the denominators to simplify your work.