Math Problem Statement
Solution
Let's analyze both systems of equations from the image.
System A:
-x + 3y - 9 &= 0 \quad \text{(Equation 1)} \\ x - 3y &= 9 \quad \text{(Equation 2)} \end{aligned}$$ **Step 1: Simplify both equations.** - From Equation 1: $$-x + 3y = 9$$. - Equation 2 remains $$x - 3y = 9$$. **Step 2: Add the equations.** $$(-x + 3y) + (x - 3y) = 9 + 9$$ This simplifies to: $$0 = 18$$ Since this is a contradiction, it means **System A has no solution**. ### System B: $$\begin{aligned} -x - 4y &= 8 \quad \text{(Equation 1)} \\ x - 4y &= 8 \quad \text{(Equation 2)} \end{aligned}$$ **Step 1: Add the two equations.** $$(-x - 4y) + (x - 4y) = 8 + 8$$ This simplifies to: $$-8y = 16$$ Solving for $$y$$: $$y = -2$$ **Step 2: Substitute $$y = -2$$ into one of the original equations, say Equation 2:** $$x - 4(-2) = 8 \quad \Rightarrow \quad x + 8 = 8 \quad \Rightarrow \quad x = 0$$ So, **System B has a unique solution**: $$(x, y) = (0, -2)$$. ### Summary: - **System A** has no solution. - **System B** has a unique solution: $$(x, y) = (0, -2)$$. Let me know if you need any more details or explanations! --- Here are 5 follow-up questions based on this: 1. How can you tell if a system of equations has no solution by looking at the equations? 2. What does it mean when a system of equations has infinitely many solutions? 3. How can you use substitution or elimination to solve a system of equations? 4. What is the geometric interpretation of a system of linear equations having no solution? 5. Can two distinct linear equations ever have exactly two solutions? **Tip:** When solving systems of equations, contradictions like $$0 = 18$$ suggest no solution, while identities like $$0 = 0$$ imply infinite solutions.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Systems of Equations
Algebra
Contradiction in Equations
Unique Solutions
Formulas
Simplification and addition of linear equations
Solving for variables in linear equations
Theorems
If two equations result in a contradiction, the system has no solution.
If two equations simplify to an identity, the system has infinitely many solutions.
Suitable Grade Level
Grades 9-12
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