Math Problem Statement
Solution
To determine which system of equations is inconsistent, we need to analyze each set of linear equations. A system is inconsistent if it has no solution, meaning the lines representing the equations are parallel and never intersect.
Let's go through each option one by one.
Option 1:
2x + 8y &= 6 \\ 5x + 20y &= 2 \end{aligned}$$ - Divide the second equation by 5: $$x + 4y = \frac{2}{5}$$ - This makes the system: $$\begin{aligned} 2x + 8y &= 6 \\ x + 4y &= \frac{2}{5} \end{aligned}$$ The ratios of the coefficients of $$x$$ and $$y$$ are proportional, but the constants (right-hand sides) are not. This means the lines are parallel but do not intersect, making this system **inconsistent**. ### Option 2: $$\begin{aligned} 5x + 4y &= -14 \\ 3x + 6y &= 6 \end{aligned}$$ This system needs further analysis: - Multiply the first equation by 3 and the second by 5: $$\begin{aligned} 15x + 12y &= -42 \\ 15x + 30y &= 30 \end{aligned}$$ The equations are not proportional, meaning the system might have a solution and is **not inconsistent**. ### Option 3: $$\begin{aligned} x + 2y &= 3 \\ 4x + 6y &= 5 \end{aligned}$$ - Multiply the first equation by 4: $$4x + 8y = 12$$ Compare this with the second equation: $$4x + 6y = 5$$ These lines are not parallel, so the system is **not inconsistent**. ### Option 4: $$\begin{aligned} 3x - 2y &= 2 \\ 6x - 4y &= 4 \end{aligned}$$ - The second equation is simply a multiple of the first equation, so these two lines are coincident, meaning they have infinitely many solutions. This system is **not inconsistent**. ### Conclusion: The system in **Option 1** is inconsistent. Would you like more details or have any questions? Here are five relative questions to expand on this: 1. How can you determine if two lines are parallel by comparing their equations? 2. What happens to a system of equations if the two lines are coincident? 3. Can a system have more than one solution, and how is this represented graphically? 4. How do you solve a system of equations using substitution or elimination? 5. What are the implications of having an inconsistent system in real-world problems? **Tip:** To check if a system of equations is inconsistent, always compare the ratios of the coefficients and constants. If the coefficients are proportional but the constants are not, the system is inconsistent.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Algebra
Linear Equations
Systems of Equations
Inconsistency
Formulas
Standard form of linear equations: Ax + By = C
Theorems
Inconsistent systems: Lines are parallel if their slopes are the same but intercepts are different
Suitable Grade Level
Grades 7-9
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