Math Problem Statement
Which system of equations has the same solution as the system below? 5x + 2y = 60, 3x + 4y = 50
Solution
The problem asks to find which system of equations among the given options has the same solution as the system:
1) \quad 5x + 2y = 60 \\ 2) \quad 3x + 4y = 50 \end{aligned}$$ ### Approach: 1. Two systems are equivalent if one can be transformed into the other by multiplying an equation by a non-zero constant or adding/subtracting multiples of the equations. 2. We analyze each option to check if it corresponds to the original system. ### Analysis: #### Option A: $$-10x - 4y = -120 \quad \text{and} \quad 3x + 4y = 50$$ - The first equation, $$-10x - 4y = -120$$, is obtained by multiplying $$5x + 2y = 60$$ by $$-2$$: $$-2(5x + 2y) = -120 \quad \Rightarrow \quad -10x - 4y = -120$$ - The second equation, $$3x + 4y = 50$$, is identical to the second equation in the original system. Thus, **Option A** is equivalent. --- #### Option B: $$-10x - 4y = 60 \quad \text{and} \quad 3x + 4y = 50$$ - The first equation, $$-10x - 4y = 60$$, cannot be obtained by scaling or manipulating $$5x + 2y = 60$$. Hence, this is **not equivalent**. --- #### Option C: $$5x - 4y = -120 \quad \text{and} \quad 3x + 4y = 50$$ - The first equation, $$5x - 4y = -120$$, cannot be derived from $$5x + 2y = 60$$. Hence, this is **not equivalent**. --- #### Option D: $$-10x + 2y = -120 \quad \text{and} \quad 3x + 4y = 50$$ - The first equation, $$-10x + 2y = -120$$, cannot be derived from $$5x + 2y = 60$$. Hence, this is **not equivalent**. --- ### Final Answer: The correct choice is **Option A**. --- Would you like a detailed explanation of how equivalency works, or have any other questions? ### Relative Questions: 1. What does it mean for two systems of equations to be equivalent? 2. How can we verify solutions for systems of linear equations? 3. Can two different systems share a solution without being equivalent? 4. How do scalar multiplications affect a system of equations? 5. How can graphing help in visualizing equivalent systems? ### Tip: When checking equivalence, focus on multiplying equations by constants or rearranging terms. These operations do not change the solutions of the system!Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Algebra
Systems of Linear Equations
Equivalent Systems
Formulas
Linear equation: ax + by = c
Theorems
Properties of Equivalent Systems
Suitable Grade Level
Grades 8-10