Math Problem Statement

Given the system of equations: 2x - 3y - 9z = -23, x + 3z = 8, and -3x + y - 4z = -11, determine the type of system (dependent or inconsistent). If dependent, express the solution in terms of z.

Solution

Let's solve this system of equations step by step to determine whether the system is dependent or inconsistent. The given system of equations is:

1) & \quad 2x - 3y - 9z = -23 \\ 2) & \quad x + 3z = 8 \\ 3) & \quad -3x + y - 4z = -11 \end{aligned}$$ ### Step 1: Solve for $$x$$ in terms of $$z$$ from the second equation. From equation (2): $$x + 3z = 8 \implies x = 8 - 3z$$ ### Step 2: Substitute $$x = 8 - 3z$$ into equations (1) and (3). #### Substitution into equation (1): $$2(8 - 3z) - 3y - 9z = -23$$ Simplifying: $$16 - 6z - 3y - 9z = -23$$ $$16 - 15z - 3y = -23$$ $$-15z - 3y = -23 - 16$$ $$-15z - 3y = -39$$ Dividing by -3: $$5z + y = 13 \quad \text{(Equation 4)}$$ #### Substitution into equation (3): $$-3(8 - 3z) + y - 4z = -11$$ Simplifying: $$-24 + 9z + y - 4z = -11$$ $$-24 + 5z + y = -11$$ $$5z + y = 13 \quad \text{(Equation 5)}$$ ### Step 3: Compare equations (4) and (5). Both equations are identical: $$5z + y = 13$$ This indicates that the system has infinitely many solutions because equations (1), (2), and (3) reduce to the same condition. This suggests that the system is **dependent**. ### Step 4: Express the solution using a free variable $$z$$. Since the system is dependent, we can express the solution in terms of $$z$$. Using $$x = 8 - 3z$$ and $$y = 13 - 5z$$, the general solution is: $$x = 8 - 3z, \quad y = 13 - 5z, \quad z = z$$ Thus, the solution is: $$(x, y, z) = (8 - 3z, 13 - 5z, z)$$ ### Final Answer: - **(a)** The system is **dependent**. - **(b)** The solution in terms of $$z$$ is $$(8 - 3z, 13 - 5z, z)$$. Would you like more details or have any questions? Here are five related questions to consider: 1. How can you tell if a system of equations is consistent or inconsistent without solving? 2. What is the difference between a dependent and independent system of equations? 3. Can all systems of linear equations be solved by substitution? If not, why? 4. What happens if a system of equations has no solutions? 5. How can you use matrix methods (like Gaussian elimination) to solve this system? **Tip:** In a dependent system, one equation is often a linear combination of the others, leading to infinitely many solutions.

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Math Problem Analysis

Mathematical Concepts

Linear Systems
Algebra
Systems of Equations

Formulas

Substitution Method
Solving Linear Equations

Theorems

Dependent and Independent Systems
Inconsistent Systems

Suitable Grade Level

Grades 10-12

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