Solve System Using Substitution

Use substitution to solve the system of equations: $y = 3x + 2$ and $y = 7x - 6$.

Solving systems of equations is a foundational skill in algebra that involves finding the values of variables that satisfy all equations in the system simultaneously. The substitution method is one of the approaches used to solve these systems, particularly when they consist of linear equations. When you use substitution, you solve one of the equations for one variable in terms of the other. Then, you substitute this expression into the other equation. This process reduces the system of equations to a single equation in one variable, which can be solved straightforwardly. In this problem, two linear equations in terms of x and y are given, and the objective is to find the point where these lines intersect, indicating the solution to the system.

Understanding the concept of substitution is valuable because it not only helps solve linear systems but also serves as a useful technique in other areas of mathematics, such as calculus, where it is used in integration by substitution. Moreover, learning how to transform and manipulate equations fosters algebraic fluency, which is critical for solving more complex mathematical problems. The ability to switch between different methods of solving systems—such as substitution, elimination, and graphical methods—offers flexibility and insight into the structure and potential solutions of algebraic equations.