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Mastering the art of solving linear systems is a fundamental skill in algebra. A linear system consists of two or more linear equations that, when graphed, create lines. The goal is to find the values that satisfy all equations simultaneously. This can be done using different methods, one of which is the substitution method.

Using substitution, you solve one equation for a variable and then replace that variable in another equation. This process leads to a single equation with one variable that can be solved. Let's look at our example: we had the system \(2x + 3y = 31\) and \(y = x + 7\). By substituting \(y\) with \(x + 7\), we could transform the system into a single equation. Sequential steps of substitution, simplification, and solving deliver an exact pair \(x, y\) that satisfies both original equations. For many students, visualizing this process as a method of 'plugging in' values can be immensely helpful.

An algebraic equation is a mathematical statement indicating that two expressions are equal. It comprises variables, constants, and arithmetic operations. In the equation \(2x + 3y = 31\), \(x\) and \(y\) are variables, while \(2\), \(3\), and \(31\) are constants. When you encounter a system of linear equations, each equation represents a piece of a puzzle that the solution will complete.

To enhance understanding, it's helpful to view each equation as a scale in balance. When you perform an action on one side, it must be mirrored on the other side to maintain the balance. Algebraic equations follow this balance concept, which is why any operation you perform to one side (like adding \(7\) to \(x\)) must also be done to the other.

Simplifying equations is the process of reducing an equation to its most basic form, making it easier to solve. This step is where we often combine like terms and move constants to one side. For example, in our system \(2x + 3(x + 7) = 31\), combining like terms \(2x\) and \(3x\) results in \(5x\).

It's important to consolidate your steps when simplifying: open the brackets, combine like terms, and isolate the variable on one side of the equation. This systematic approach not only leads to a correctly solved equation but also helps avoid errors. Additionally, double-checking each step can ensure accuracy before moving on to the next part of the problem.

A system of equations represents a collection of two or more equations with a set of variables. The aim is to find a common solution that satisfies all equations within the system. The two main types of solutions to a system are the intersecting point, where two lines cross and provide a unique solution, and the overlapping lines, where infinite solutions exist. Another possibility is parallel lines, which mean there is no solution since the lines never meet.

In the provided exercise, the system is composed of two equations: \(2x + 3y = 31\) and \(y = x + 7\). Here, we are looking for the intersecting point of the lines represented by these equations. By following the substitution method, we can find the precise point of intersection and provide the unique solution to the system.