If a system has an infinite number of solutions, use...

If a system has an infinite number of solutions, use set-builder notation to write the solution set. If a system has no solution, state this.
Solve using the substitution method.
$$
\begin{aligned}
&4 x-2 y=6\\
&2 x-3=y
\end{aligned}
$$

Key Concepts

Set-Builder Notation

Set-builder notation is a method used to describe a set by specifying the properties that its members must satisfy. It is especially useful for expressing the solution set of a system with infinitely many solutions by clearly indicating the relationship between the variables, such as describing solutions as all ordered pairs that satisfy a given condition.

Substitution Method

The substitution method is a way to solve a system of equations by expressing one variable in terms of another using one equation, and then substituting this expression into the other equation(s). This technique simplifies the system, reducing it to an equation with a single variable, which makes it easier to solve.

Infinite Solution Set

An infinite solution set occurs when the system of equations is dependent, meaning that one equation is effectively a multiple of the other, so they represent the same line or plane. In such cases, there are infinitely many solutions because every solution of one equation automatically satisfies the other.

System of Linear Equations

A system of linear equations consists of two or more equations with linear expressions. These systems can have a unique solution, infinitely many solutions, or no solution, depending on whether the lines or planes represented by the equations intersect at a point, coincide, or are parallel and distinct.

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