10. \( \begin{aligned} 2 x-5 & =-y \\ x+3 y & =0\end{aligned} \)

To solve the system of equations \( 2x - 5 = -y \) and \( x + 3y = 0 \), we can substitute \( y \) from the first equation into the second. Rearranging the first equation gives us \( y = -2x + 5 \). Substituting this into the second equation: \[ x + 3(-2x + 5) = 0 \] Now combine like terms: \[ x - 6x + 15 = 0 \] which simplifies to \[ -5x + 15 = 0 \] Solving for \( x \), we find \[ -5x = -15 \quad \Rightarrow \quad x = 3 \] Next, we substitute \( x = 3 \) back into \( y = -2x + 5 \): \[ y = -2(3) + 5 = -6 + 5 = -1 \] So, the solution to the system of equations is \( (3, -1) \). --- Did you know that systems of equations were first discussed systematically by mathematicians like René Descartes in the 17th century? Descartes introduced a coordinate system that allows for the graphical representation of equations, making it easier for later scholars to solve these systems visually! In the real world, systems of equations are incredibly useful. They're used in various fields, including economics to analyze market behaviors, in engineering for design optimizations, and in budgeting to allocate resources efficiently. Knowing how to solve these systems can help you tackle everything from personal finance to complex project planning!