Solve the system. If the system has one unique solution, write the solution set. Otherwise, determine the number of solutions to the system, whether the system is inconsistent, or the equations are dependent.
[tex]
\begin{array}{rrl}
-3 x-4 y & +4 z & =-3 \\
2 y-z & =1 \\
-3 x & +2 z & =-1
\end{array}
[/tex]
The system has one solution.
The solution set is {([ ])}
The system has no solution.
The system is inconsistent.
The equations are dependent.
The system has infinitely many solutions.
The system is inconsistent.
The equations are dependent.
Answer (2)
Express z in terms of y using the second equation: z = 2 y − 1 .
Substitute z into the first and third equations to obtain two equations in terms of x and y .
Observe that the two equations are identical, indicating that the system has infinitely many solutions and the equations are dependent.
Express the solution set in terms of a parameter, such as y : { ( 3 4 y − 1 , y , 2 y − 1 ) } .
Explanation
Analyzing the Problem We are given a system of three linear equations with three variables x , y , and z :
− 3 x − 4 y + 4 z 2 y − z − 3 x + 2 z = − 3 = 1 = − 1
Our goal is to solve this system and determine whether it has a unique solution, no solution, or infinitely many solutions. If a unique solution exists, we need to find the values of x , y , and z .
Expressing z in terms of y From the second equation, we can express z in terms of y :
z = 2 y − 1
Substituting this expression for z into the third equation, we get:
− 3 x + 2 ( 2 y − 1 ) = − 1
Simplifying this equation, we have:
− 3 x + 4 y − 2 = − 1
− 3 x + 4 y = 1
Substituting z into the first equation Now, let's substitute z = 2 y − 1 into the first equation:
− 3 x − 4 y + 4 ( 2 y − 1 ) = − 3
Simplifying this equation, we get:
− 3 x − 4 y + 8 y − 4 = − 3
− 3 x + 4 y = 1
Notice that this equation is the same as the one we obtained from substituting into the third equation. This indicates that the equations are dependent, and the system has infinitely many solutions.
Finding the Solution Set Since we have − 3 x + 4 y = 1 , we can solve for x in terms of y :
− 3 x = 1 − 4 y
x = 3 4 y − 1
Also, we have z = 2 y − 1 . Thus, the solution set can be expressed in terms of the parameter y as:
{ ( 3 4 y − 1 , y , 2 y − 1 ) }
This means that for any value of y , we can find corresponding values of x and z that satisfy the system of equations.
Expressing the Solution in Different Ways Alternatively, we can express the solution in terms of z . From z = 2 y − 1 , we have y = 2 z + 1 . Substituting this into x = 3 4 y − 1 , we get:
x = 3 4 ( 2 z + 1 ) − 1 = 3 2 z + 2 − 1 = 3 2 z + 1
So the solution set can also be expressed in terms of z as:
{ ( 3 2 z + 1 , 2 z + 1 , z ) }
Similarly, we can express the solution in terms of x . From − 3 x + 4 y = 1 , we have y = 4 3 x + 1 . Substituting this into z = 2 y − 1 , we get:
z = 2 ( 4 3 x + 1 ) − 1 = 2 3 x + 1 − 1 = 2 3 x − 1
So the solution set can also be expressed in terms of x as:
{ ( x , 4 3 x + 1 , 2 3 x − 1 ) }
Since there are infinitely many solutions, the system is not inconsistent, and the equations are dependent.
Final Answer The system has infinitely many solutions, and the equations are dependent. The solution set can be expressed as { ( 3 4 y − 1 , y , 2 y − 1 ) } .
Examples
Systems of linear equations are used in various real-world applications, such as modeling electrical circuits, balancing chemical equations, and determining the optimal mix of ingredients in a recipe. For instance, consider a nutritionist who wants to create a meal plan with specific amounts of carbohydrates, proteins, and fats. By setting up a system of linear equations, the nutritionist can determine the quantities of different foods needed to meet the desired nutritional targets. Similarly, engineers use systems of equations to analyze the forces acting on a structure and ensure its stability.
The system of equations has infinitely many solutions, as the dependent equations produce a parameterized solution set. The solution set can be described as {(\frac{4y - 1}{3}, y, 2y - 1)}, meaning any value of y corresponds to specific x and z values. Therefore, the equations are not inconsistent, and there is no unique solution.
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