91Ó°ÊÓ

When tackling a problem involving a system of equations, you're working with two or more equations that share the same set of unknowns, usually written in terms of variables like \(x\) and \(y\). The goal is to find values for these variables that satisfy all of the given equations simultaneously. For instance, let's consider the provided system: \[\left\{\begin{array}{l}y = 25 + 30x \y = 15 + 32x\end{array}\right.\]Both equations relate the variables \(x\) and \(y\). To solve this, we want to find a single pair of values that make both equations true. Methods such as graphing, elimination, and substitution can help us find this solution. In our exercise, we've applied the substitution method, which involves expressing one variable in terms of the other using one of the equations, and then replacing this expression in the other equation.

Algebra is fundamentally about finding the unknown. This makes it an essential tool for solving systems of equations. The substitution method we used is a staple technique in algebra. It relies on manipulating algebraic expressions to express one variable in terms of another.

In our exercise, both equations were already solved for \(y\). This means they were both given as functions of \(x\). We started by setting the expressions for \(y\) equal to each other: \[25 + 30x = 15 + 32x\]This step allowed us to simplify and consolidate our problem into a single equation with one variable, \(x\). By performing algebraic operations like subtracting and simplifying, we reached the result \(x = 5\). Once we know the value of one variable, we can substitute it back into one of the original equations to find the other variable.

Verification is an integral part of solving equations. Once you find a solution, you need to check that it satisfies all of the original equations in the system. This helps confirm the accuracy of your calculations and ensures no mistakes were made.To verify our solution, \(x = 5\) and \(y = 175\), we substituted \(x\) back into both original equations:1. First equation: \( y = 25 + 30(5) = 175 \) 2. Second equation: \( y = 15 + 32(5) = 175 \)Since both equations are satisfied, we are confident that our solution is correct. This step is crucial and should always be performed to validate your work, ensuring that every aspect of the original problem is addressed.