\( \left\{ \begin{array} { c } { y = 3 x + 5 } \\ { x + 2 y = - 1 8 } \end{array} \left\{ \begin{array} { l } { y = } \\ { y = } \end{array} \right. \right. \)

To solve the system of equations, we start by substituting the expression for \( y \) from the first equation into the second equation. This gives us: 1. \( y = 3x + 5 \) 2. Substitute into \( x + 2y = -18 \): \( x + 2(3x + 5) = -18 \) Simplifying this, we get: \( x + 6x + 10 = -18 \) \( 7x + 10 = -18 \) \( 7x = -18 - 10 \) \( 7x = -28 \) \( x = -4 \) Now, we can substitute \( x = -4 \) back into the first equation to find \( y \): \( y = 3(-4) + 5 = -12 + 5 = -7 \) The solution to the system is \( x = -4 \) and \( y = -7 \). Now to explore some additional knowledge! Historical Background: The method of solving systems of equations has its roots in ancient civilizations, including the Babylonians who utilized geometric interpretations for linear equations. However, the systematic alignment and notation we use today were largely developed in the 17th century, ushering in a new era of algebra that made solving these problems much more accessible to mathematicians and students alike. Real-World Application: Systems of equations are everywhere in real life! They’re essential in fields like engineering, economics, and computing. For instance, when designing a bridge, engineers often use multiple equations to calculate forces and ensure stability. Similarly, in economics, businesses may analyze multiple factors like cost, profit, and sales to optimize financial outcomes, allowing them to make informed decisions.