1. **State the problem:** Solve the system of linear equations: $$4x + y = 7$$ $$5x + 3y = 7$$ 2. **Formula and method:** We can use the substitution or elimination method. Here, we'll use elimination. 3. **Elimination method:** Multiply the first equation by 3 to align the coefficients of $y$: $$3(4x + y) = 3(7) \Rightarrow 12x + 3y = 21$$ 4. **Subtract the second equation from this new equation:** $$ (12x + 3y) - (5x + 3y) = 21 - 7 $$ $$ 12x - 5x + 3y - 3y = 14 $$ $$ 7x = 14 $$ 5. **Solve for $x$:** $$ x = \frac{14}{7} = 2 $$ 6. **Substitute $x=2$ into the first original equation to find $y$:** $$ 4(2) + y = 7 $$ $$ 8 + y = 7 $$ $$ y = 7 - 8 = -1 $$ 7. **Final answer:** $$ x = 2, \quad y = -1 $$