Math Problem Statement

Solve the system of equations by Elimination/Addition: -4x + 4y = 28 -6x + 6y = 43

Solution

The system of equations provided is:

-4x + 4y &= 28 \\ -6x + 6y &= 43 \end{aligned}$$ Let's solve this system by elimination or substitution. ### Step 1: Simplify the equations if possible Notice both equations share common factors: 1st equation: $$-4x + 4y = 28 \quad \text{can be simplified to} \quad -x + y = 7$$ 2nd equation: $$-6x + 6y = 43 \quad \text{cannot be simplified easily}$$ ### Step 2: Eliminate one variable We can multiply the first equation by 6 and the second by 4 to align the coefficients of $$x$$: $$6(-x + y) = 6(7) \quad \Rightarrow \quad -6x + 6y = 42$$ Now subtract this from the second equation: $$(-6x + 6y) - (-6x + 6y) = 43 - 42$$ This simplifies to: $$0 = 1$$ ### Step 3: Analyze the result The result $$0 = 1$$ is a contradiction, meaning the system of equations has no solution. This happens when the lines represented by the equations are parallel and never intersect. ### Final Answer: There is **no solution** for this system. Would you like more details or have any questions? Here are 5 related questions you might want to explore: 1. How do you identify if a system of equations is inconsistent? 2. What are the steps for solving systems of equations by substitution? 3. What is the geometric interpretation of parallel lines in a system of equations? 4. How can the determinant of a matrix help determine if a system has a solution? 5. What is the difference between consistent and inconsistent systems of equations? **Tip:** Always check if simplifying the system at the start can make solving easier.

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Math Problem Analysis

Mathematical Concepts

Linear Algebra
Systems of Linear Equations

Formulas

Elimination method
Addition method

Theorems

System of equations consistency theorem

Suitable Grade Level

Grades 9-12

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