Question

solve the system of equations: \\(\

$$\begin{cases} 2x - 3y = 9 \\\\ -5x - 3y = 30 \\end{cases}$$

\\)

Explanation:

Step1: Subtract the two equations

We have the system of equations:

$$\begin{cases} 2x - 3y = 9 \\ -5x - 3y = 30 \end{cases}$$

Subtract the first equation from the second equation (or vice versa) to eliminate \(y\). Let's subtract the first equation from the second:
\((-5x - 3y)-(2x - 3y)=30 - 9\)
Simplify the left - hand side: \(-5x-3y - 2x + 3y=-7x\)
Simplify the right - hand side: \(30 - 9 = 21\)
So we get the equation \(-7x=21\)

Step2: Solve for \(x\)

From \(-7x = 21\), divide both sides by \(-7\):
\(x=\frac{21}{-7}=-3\)

Step3: Substitute \(x = - 3\) into one of the original equations to solve for \(y\)

Let's substitute \(x=-3\) into the first equation \(2x-3y = 9\)
We have \(2\times(-3)-3y=9\)
Simplify the left - hand side: \(-6-3y = 9\)
Add 6 to both sides: \(-3y=9 + 6=15\)
Divide both sides by \(-3\): \(y=\frac{15}{-3}=-5\)

Answer:

The solution to the system of equations is \(x=-3,y = - 5\) or \((-3,-5)\)