\( \left. \begin{array} { r } { 10 s + 7 t = 69 } \\ { 4 s + 5 t = 43 } \\ { s = 2 , t = 7 } \end{array} \right. \)
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The Deep Dive
Did you know that linear equations like these are the building blocks of algebra? This system of equations shows us how two variables, \( s \) and \( t \), can work together to find a solution that satisfies both equations. Solving such systems can help model real-world scenarios, like budgeting or resource allocation, where you need to find the best combination of different items or services! When trying to solve systems of equations, a common mistake is to misinterpret the coefficients or constant terms. Always double-check your calculations after substituting values back into the original equations to ensure they hold true. Keeping track of signs (positive/negative) is crucial, especially when dealing with multiple variables!