A Step-by-Step Guide to Finding a Parametric Description of the Solution Set of a Linear System

Finding a parametric description of the solution set of a linear system is the same as solving the system

To find a parametric description of the solution set of a linear system, we first need to solve the system

To find a parametric description of the solution set of a linear system, we first need to solve the system. Solving the linear system involves finding the values of the variables that make all of the equations in the system true simultaneously.

Once we have found the solution to the system, we can then express the solution set using parametric descriptions. Parametric descriptions involve expressing the variables in terms of a parameter (usually denoted by a letter such as “t” or “s”). This allows us to describe the infinite number of solutions of a system by using a finite number of equations.

Here is a step-by-step approach to finding a parametric description of the solution set of a linear system:

1. Write down the given system of equations. For example, consider the following system:
Equation 1: 2x + y = 3
Equation 2: x – 3y = 1

2. Solve the system by using any appropriate method (e.g., substitution, elimination, or matrix methods). In this case, we can solve the system using the method of elimination:
Multiply Equation 1 by 3: 6x + 3y = 9
Add Equation 2 to the modified Equation 1: 7x = 10
Solve for x: x = 10/7

Substitute the value of x into either original equation to solve for y:
Using Equation 1: 2(10/7) + y = 3
Simplify and solve for y: y = 11/7

Therefore, the solution to the system is x = 10/7 and y = 11/7.

3. Express the solution set using parametric descriptions. Since we have only two variables, we can express one variable in terms of the parameter and use the other variable as a free variable. Let’s express x in terms of t, where t is the parameter:
x = 10/7 + t

Now, express y in terms of t:
y = 11/7

The parametric description of the solution set is:
x = 10/7 + t
y = 11/7

4. Verify the parametric description. Substitute different values of the parameter t into the equations and check if they satisfy the original system. For example, let’s substitute t = 0 into the parametric description and check if we get the original equations:
x = 10/7 + 0 = 10/7
y = 11/7

Plugging these values into the original equations:
Equation 1: 2(10/7) + (11/7) = 20/7 + 11/7 = 31/7 = 3 (True)
Equation 2: (10/7) – 3(11/7) = 10/7 – 33/7 = -23/7 = 1 (True)

Since both equations are true, we have verified that the parametric description is correct.

In summary, finding a parametric description of the solution set of a linear system involves first solving the system using appropriate methods. After finding the solution, we can express the solution set using parametric descriptions by expressing the variables in terms of a parameter.

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