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Algebraic methods are techniques used to solve equations in algebra. These methods help find the values of variables in mathematical expressions. They come in handy for simplifying complex equations and systems of equations. Understanding algebraic methods starts with knowing the properties of algebra, such as addition, subtraction, multiplication, and division of expressions and equations. These basic functions allow you to manipulate equations effectively. One common approach is simplifying equations by dividing or multiplying each term by a specific number. In the exercise provided, algebraic methods involve simplifying each equation by dividing by 10. This step makes the numbers more manageable and sets the stage for solving the system. Simplifying allows us to see relationships within the equations clearer, making further steps like elimination method easier to perform. Remember, correctly applying algebraic methods is crucial for efficiently solving any system of linear equations.

The elimination method is a powerful tool when solving systems of linear equations. It involves adjusting equations so that adding or subtracting them removes one of the variables. This reduction simplifies the system, allowing us to solve for each variable one at a time.In our exercise, the elimination method was particularly useful. After simplifying both equations, the next step was to manipulate one equation to align coefficients for one variable. Here, multiplying the second equation by 2 allowed the coefficients of the variable \( s \) to match.Once matched, subtracting one equation from the other removed \( s \), leaving an equation with a single variable \( t \). This simplification made it easy to solve for \( t \), which then allows substitution back into an original equation to find \( s \).Using the elimination method helps solve systems efficiently. It’s especially helpful when straightforward methods, like direct substitution, become too cumbersome due to many variables or complex coefficients.

Solving systems of equations is about finding values for each variable that satisfy all equations involved. Systems of linear equations can be tricky, but methods like substitution, elimination, and graphical methods simplify the process.In the given problem, we aimed to find the values of \( s \) and \( t \) that satisfy both equations simultaneously. We began by simplifying the system using division algebraically to reduce complexity.After simplifying, the elimination method came into play to remove one of the variables and focus on solving one equation. By solving for one variable first, then substituting back into one of the original equations, we found the complete set of solutions. The solution \( (s, t) = \left( \frac{39}{10}, \frac{16}{5} \right) \) was achieved through careful application of both simplification and elimination. Mastering solving systems of equations requires practice and the ability to choose the right method based on the given equations for efficient problem-solving.