Solve the system using the substitution method. For systems...

Solve the system using the substitution method. For systems that do not have one unique solution, also state the number of solutions and whether the system is inconsistent or the equations are dependent.
$$\begin{aligned}&y=6.8 x+2.3\\&y=-4.1 x+56.8\end{aligned}$$

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Key Concepts

Unique, Dependent, and Inconsistent Systems

When solving a system of linear equations, it is important to determine the nature of its solutions. A unique solution occurs when the equations intersect at exactly one point. A dependent system has infinitely many solutions because the equations represent the same line, while an inconsistent system has no solution because the equations represent parallel lines with no point of intersection.

Substitution Method

The substitution method is a technique used to solve systems of equations where one equation is solved for one variable and the resulting expression is substituted into the other equation. This process reduces the system of equations to one with a single variable, making it easier to solve. Once the single variable is determined, it is substituted back to find the other variable.

Systems of Linear Equations

A system of linear equations consists of two or more equations involving the same set of variables. The goal is to find the values of the variables that simultaneously satisfy all the equations in the system. These systems can be solved using various methods, including substitution, elimination, or matrix approaches.

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