Use the substitution method to find all solutions of each system. { 4 x+6 y =3 -6 x-9 y

step 1 Here's a system of equations that we're going to solve using the substitution method. And in nei

Use the substitution method to find all solutions of each...

Key Concepts

System of Linear Equations

This concept involves working with two or more equations that contain the same set of variables, with the goal of finding values for these variables that satisfy all equations simultaneously. Understanding how the equations interact and relate to each other is crucial for determining whether a unique solution, infinitely many solutions, or no solution exists.

Substitution Method

The substitution method is a procedure for solving systems of equations where one equation is solved for one variable, which is then substituted into the other equation(s). This method reduces the system step by step and is especially useful when one of the equations is easily solvable for one of the variables.

Linear Dependency and Consistency

In the context of systems of linear equations, linear dependency occurs when one equation is a scalar multiple of another, leading to either infinitely many solutions or, in some cases, no solution if the equations are inconsistent. Recognizing linear dependency helps in determining whether the system is consistent (has at least one solution) or inconsistent (has no solution).

Infinite Solutions

This concept arises when the equations in a system are not independent but rather represent the same line or plane. In such cases, after applying methods like substitution, the result is a parametric equation that describes infinitely many solutions, reflecting the fact that there is more than one way to satisfy all the original equations simultaneously.

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