Solve the system. If the system has one unique solution, write the solution set. Otherwise, determine the number of solutions to the system, and determine whether the system is inconsistent, or the equations are dependent.
[tex]
\begin{aligned}
-3 x-4 y+4 z & =-3 \\
2 y-z & =1 \\
-3 x & +2 z\end{aligned}=-1
[/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 eliminate z , resulting in − 3 x + 4 y = 1 for both.
Solve for x in terms of y : x = 3 4 y − 1 .
Since y is a free variable ( y = t ), the system has infinitely many solutions, and the equations are dependent. The solution set is x = 3 4 t − 1 , y = t , z = 2 t − 1 .
Explanation
Analyzing the System of Equations We are given a system of three linear equations in three variables x, y, and z:
-3x - 4y + 4z = -3
2y - z = 1
-3x + 2z = -1
Our objective is to solve the system of equations and determine if there is a unique solution, no solution (inconsistent), or infinitely many solutions (dependent). If there is a unique solution, we need to find the solution set.
Expressing z in terms of y From the second equation, we can express z in terms of y:
z = 2 y − 1
Substitute this expression for z into the third equation to eliminate z:
− 3 x + 2 ( 2 y − 1 ) = − 1
Simplify the equation:
− 3 x + 4 y − 2 = − 1
− 3 x + 4 y = 1
Now we have an equation relating x and y.
Substituting z into the first equation Substitute the expression for z, z = 2 y − 1 , into the first equation:
− 3 x − 4 y + 4 ( 2 y − 1 ) = − 3
Simplify the equation:
− 3 x − 4 y + 8 y − 4 = − 3
− 3 x + 4 y = 1
Notice that this equation is the same as the one we derived from the third equation.
Solving for x and introducing a free variable Since the first and third equations (after substitution) are identical, we have only two independent equations. This indicates that the system has either no solution or infinitely many solutions. Let's solve the equation − 3 x + 4 y = 1 for x in terms of y:
3 x = 4 y − 1
x = 3 4 y − 1
Let y be a free variable, say y = t . Then x = 3 4 t − 1 and z = 2 t − 1 .
Verifying the Solution To verify the solution, substitute x = 3 4 t − 1 , y = t , and z = 2 t − 1 into the original equations:
-3x - 4y + 4z = -3
− 3 ( 3 4 t − 1 ) − 4 t + 4 ( 2 t − 1 ) = − ( 4 t − 1 ) − 4 t + 8 t − 4 = − 4 t + 1 − 4 t + 8 t − 4 = − 3
2y - z = 1
2 t − ( 2 t − 1 ) = 2 t − 2 t + 1 = 1
-3x + 2z = -1
− 3 ( 3 4 t − 1 ) + 2 ( 2 t − 1 ) = − ( 4 t − 1 ) + 4 t − 2 = − 4 t + 1 + 4 t − 2 = − 1
The solution satisfies all three original equations. Since we have a free variable, the system has infinitely many solutions. The equations are dependent.
Final Answer The system has infinitely many solutions, and the equations are dependent. The solution set can be expressed as:
x = 3 4 t − 1 , y = t , z = 2 t − 1
where t is any real number.
Examples
Systems of equations are used in various real-world applications, such as modeling electrical circuits, analyzing supply and demand in economics, and optimizing resource allocation in engineering. For instance, consider a scenario where a company produces three different products, each requiring different amounts of labor, materials, and capital. By setting up a system of equations, the company can determine the optimal production levels for each product to maximize profit while satisfying constraints on available resources. Understanding how to solve systems of equations is crucial for making informed decisions in many fields.
The system of equations has infinitely many solutions and the equations are dependent. The solution set can be expressed as (x, y, z) = (\frac{4t - 1}{3}, t, 2t - 1) where t is any real number. Each variable is linked to a free parameter, indicating infinite possibilities for solutions.
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