Question

1. 7x + 12y + 25 = 0

25 = 4y + 9x

Explanation:

Step1: Rearrange the second equation

We have the second equation \(25 = - 4y+9x\), we can rewrite it as \(9x - 4y-25 = 0\) and the first equation is \(7x + 12y+25 = 0\). Now we have a system of linear equations:
\(

$$\begin{cases}7x + 12y=- 25&(1)\\9x-4y = 25&(2)\end{cases}$$

\)

Step2: Eliminate \(y\)

Multiply equation \((2)\) by \(3\) to make the coefficients of \(y\) opposite.
\(3\times(9x - 4y)=3\times25\), which gives \(27x-12y = 75\ (3)\)

Step3: Add equation \((1)\) and \((3)\)

Add \((1)\) and \((3)\): \((7x + 12y)+(27x-12y)=-25 + 75\)
Simplify the left - hand side: \(7x+27x+12y - 12y=34x\)
Simplify the right - hand side: \(50\)
So \(34x=50\), then \(x=\frac{50}{34}=\frac{25}{17}\)

Step4: Substitute \(x\) into equation \((2)\)

Substitute \(x = \frac{25}{17}\) into \(9x-4y = 25\)
\(9\times\frac{25}{17}-4y=25\)
\(\frac{225}{17}-4y=25\)
\(-4y=25-\frac{225}{17}\)
\(25=\frac{425}{17}\), so \(-4y=\frac{425 - 225}{17}=\frac{200}{17}\)
\(y=\frac{\frac{200}{17}}{-4}=-\frac{50}{17}\)

Answer:

\(x = \frac{25}{17},y=-\frac{50}{17}\)