Math Problem Statement

Determine whether the following system of linear equations is independent, inconsistent, or dependent: System 1: 2x + y = 3 4x + 2y = 6 System 2: 2x + 3y = 1 4x + 6y = 2

Solution

Let's analyze the systems of equations to determine whether they are independent, inconsistent, or dependent.

First System:

2x + y = 3 \\ 4x + 2y = 6 \end{cases}$$ 1. Multiply the first equation by 2 to see if it's proportional to the second equation: $$2(2x + y) = 2 \cdot 3 \quad \Rightarrow \quad 4x + 2y = 6$$ This shows that both equations are equivalent, meaning the system represents the same line. Therefore, the system is **dependent** (infinite solutions). ### Second System: $$\begin{cases} 2x + 3y = 1 \\ 4x + 6y = 2 \end{cases}$$ 2. Again, multiply the first equation by 2: $$2(2x + 3y) = 2 \cdot 1 \quad \Rightarrow \quad 4x + 6y = 2$$ This shows that the second equation is also a multiple of the first. Thus, this system is **dependent** as well (infinite solutions). ### Answers: 1. **Dependent** 2. **Dependent** Would you like further details or have any other questions? ### 5 Related Questions: 1. How can you graphically interpret dependent systems of equations? 2. What is the difference between inconsistent and independent systems of equations? 3. How do you solve a system of linear equations using substitution or elimination? 4. What conditions make a system of linear equations inconsistent? 5. How can you identify if a system has a unique solution? ### Tip: When the equations in a system are proportional (one is a multiple of the other), the system is dependent, meaning the two equations represent the same line.

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Math Problem Analysis

Mathematical Concepts

Linear Equations
Systems of Equations
Proportional Relationships

Formulas

Standard form of a linear equation: Ax + By = C

Theorems

Dependent Systems: Equations are multiples of each other, representing the same line.

Suitable Grade Level

Grades 9-12

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