1. **Stating the problem:** Solve the system of linear equations: $$5x - 2y = 1$$ $$4x + 3y = -10.7$$ 2. **Formula and rules:** To solve a system of two linear equations, we can use substitution or elimination. Here, elimination is convenient. 3. **Step 1: Multiply equations to align coefficients for elimination.** Multiply the first equation by 3 and the second by 2 to align $y$ coefficients: $$3(5x - 2y) = 3(1) \Rightarrow 15x - 6y = 3$$ $$2(4x + 3y) = 2(-10.7) \Rightarrow 8x + 6y = -21.4$$ 4. **Step 2: Add the two equations to eliminate $y$:** $$15x - 6y + 8x + 6y = 3 - 21.4$$ $$23x = -18.4$$ 5. **Step 3: Solve for $x$:** $$x = \frac{-18.4}{23} = -0.8$$ 6. **Step 4: Substitute $x = -0.8$ into one original equation to find $y$.** Using the first equation: $$5(-0.8) - 2y = 1$$ $$-4 - 2y = 1$$ $$-2y = 1 + 4 = 5$$ $$y = \frac{-5}{2} = -2.5$$ 7. **Final answer:** $$x = -0.8, \quad y = -2.5$$