Question

1. $-24 - 8x = 12y$

$1 + \frac{5}{9}y = -\frac{7}{18}x$

Explanation:

Step1: Simplify the first equation

Start with \(-24 - 8x = 12y\). Divide every term by 4 to simplify: \(\frac{-24}{4}-\frac{8x}{4}=\frac{12y}{4}\), which gives \(-6 - 2x = 3y\). Rearrange to standard form: \(2x + 3y = -6\). Let's call this Equation (1).

Step2: Simplify the second equation

Take \(1+\frac{5}{9}y = -\frac{7}{18}x\). Multiply every term by 18 to eliminate denominators: \(18\times1 + 18\times\frac{5}{9}y = 18\times(-\frac{7}{18}x)\). Calculate each term: \(18 + 10y = -7x\). Rearrange to standard form: \(7x + 10y = -18\). Let's call this Equation (2).

Step3: Solve the system of equations

We have:
Equation (1): \(2x + 3y = -6\)
Equation (2): \(7x + 10y = -18\)

Let's use the elimination method. Multiply Equation (1) by 10: \(20x + 30y = -60\) (Equation 3)
Multiply Equation (2) by 3: \(21x + 30y = -54\) (Equation 4)

Subtract Equation (3) from Equation (4): \((21x + 30y)-(20x + 30y)= -54 - (-60)\)
Simplify: \(x + 0y = 6\), so \(x = 6\).

Step4: Substitute \(x = 6\) into Equation (1)

Substitute \(x = 6\) into \(2x + 3y = -6\): \(2(6)+3y = -6\)
Calculate: \(12 + 3y = -6\)
Subtract 12 from both sides: \(3y = -6 - 12 = -18\)
Divide by 3: \(y = -6\).

Answer:

The solution to the system is \(x = 6\), \(y = -6\)