1. **State the problem:** Solve the system of linear equations: $$4x + 5y = 6$$ $$7x + 9y = 12$$ 2. **Formula and method:** We can use the method of substitution or elimination. Here, we'll use elimination. 3. **Elimination method:** Multiply the first equation by 7 and the second by 4 to align coefficients of $x$: $$7(4x + 5y) = 7 \times 6 \Rightarrow 28x + 35y = 42$$ $$4(7x + 9y) = 4 \times 12 \Rightarrow 28x + 36y = 48$$ 4. **Subtract the first new equation from the second:** $$ (28x + 36y) - (28x + 35y) = 48 - 42 \Rightarrow y = 6$$ 5. **Substitute $y=6$ back into the first original equation:** $$4x + 5(6) = 6 \Rightarrow 4x + 30 = 6$$ $$4x = 6 - 30 = -24$$ $$x = \frac{-24}{4} = -6$$ 6. **Final answer:** $$x = -6, \quad y = 6$$