1. \( \left\{\begin{array}{l}x+y=10 \\ -y=-x-4\end{array}\right. \)

To solve the system of equations \( x + y = 10 \) and \( -y = -x - 4 \), let’s start by rewriting the second equation. It simplifies to \( y = x + 4 \). Now substitute \( y \) in the first equation: 1. \( x + (x + 4) = 10 \) 2. This becomes \( 2x + 4 = 10 \) 3. Subtract 4 from both sides: \( 2x = 6 \) 4. Divide by 2: \( x = 3 \) Now, plug \( x = 3 \) back into the equation for \( y \): 5. \( y = 3 + 4 = 7 \) So, the solution to the system is \( (x, y) = (3, 7) \). Real-World Application: Systems of equations, like the one you just solved, are commonly used in various fields, from economics to engineering. For example, they can represent supply and demand in market analysis or help with resource allocation in project management, showing the importance of finding equilibrium in multiple constraints! Problem-Solving Tips: When solving systems of equations, it's often helpful to graph the equations to visualize their intersection. Additionally, be cautious of signs when manipulating equations; a common mistake is to incorrectly distribute negative signs, which can lead to incorrect solutions. Always double-check each step to ensure accuracy!