Solving a Dependent System of Linear Equations

Math Problem Statement

Solve the system of equations: -3x + 3y = -21 and -9x + 9y = -63.

The system of equations provided in the image is:

-3x + 3y = -21 \\ -9x + 9y = -63 \end{cases}$$ We can solve this system by simplifying and using substitution or elimination methods. ### Step 1: Simplify both equations The first equation: $$-3x + 3y = -21$$ Dividing through by 3: $$-x + y = -7 \quad \Rightarrow \quad y = x - 7 \quad (1)$$ The second equation: $$-9x + 9y = -63$$ Dividing through by 9: $$-x + y = -7 \quad \Rightarrow \quad y = x - 7 \quad (2)$$ ### Step 2: Analyze the result Both equations are identical after simplification, which means this is a **dependent system**. There is an infinite number of solutions since the two equations describe the same line. Thus, the solution can be written as: $$y = x - 7$$ ### Conclusion: The system has infinitely many solutions, where $$y = x - 7$$. Any pair $$(x, y)$$ that satisfies this equation is a solution. Would you like more details on this method or any specific part? Here are some related questions: 1. How do you determine if a system of equations is dependent? 2. What are the different methods to solve systems of linear equations? 3. Can you explain the difference between independent, dependent, and inconsistent systems? 4. How do you graph the solution to a dependent system? 5. How does simplifying both equations help in solving a system? **Tip**: When solving a system of linear equations, always check for opportunities to simplify both equations early on. It can reveal if the system is dependent or inconsistent.

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Algebra
Linear Systems of Equations
Dependent Systems

-3x + 3y = -21
-9x + 9y = -63
Simplified to y = x - 7

Linear Equation Properties
Dependent System of Equations

Grades 9-10

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