$$5x + 3y = 9 \quad \text{(I)}$$ $$2x - 3y = 12 \quad \text{(II)}$$ (linear equations, system of equations, variables, coefficients)

Step 1: Add the two equations

We have a system of two linear equations. Notice that the coefficients of $y$ are $+3$ and $-3$. Adding the two equations together will eliminate the $y$ variable, allowing us to solve for $x$.

Step 2: Simplify and solve for x

After adding the equations, the $3y$ and $-3y$ terms cancel out, leaving us with $7x = 21$. Dividing both sides by 7 gives us the value of $x$.

Step 3: Substitute x into one of the original equations

Now that we have the value of $x$, we can substitute it into either of the original equations to find the value of $y$. Let's use Equation (I): $5x + 3y = 9$.

Step 4: Solve for y

Substitute $x=3$ into the equation $5x + 3y = 9$. This simplifies to $15 + 3y = 9$. Subtract 15 from both sides to get $3y = -6$, and then divide by 3 to find $y = -2$.

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