Math Problem Statement

Use the method of elimination to solve the following system of equations. If the system is dependent, express the solution set in terms of one of the variables. Leave all fractional answers in fraction form. The system is: -2x + 5y = 26 and -10x + 25y = 130.

Solution

The system of equations provided in the image is:

-2x + 5y &= 26 \\ -10x + 25y &= 130 \end{aligned}$$ We are asked to solve this system using the **method of elimination** and express the solution in fractional form if necessary. ### Step 1: Simplify the second equation Notice that the second equation is a multiple of the first equation. We can divide the second equation by 5 to simplify: $$\frac{-10x + 25y}{5} = \frac{130}{5}$$ $$-2x + 5y = 26$$ Now both equations are identical: $$-2x + 5y = 26$$ $$-2x + 5y = 26$$ ### Step 2: Analyze the system Since both equations are the same, this indicates that the system is **dependent**. Therefore, the solution set consists of infinitely many solutions. ### Step 3: Express the solution in terms of one variable We can solve for $$x$$ in terms of $$y$$ (or vice versa) using the first equation. From the equation $$-2x + 5y = 26$$, solve for $$x$$: $$-2x = 26 - 5y$$ $$x = \frac{26 - 5y}{2}$$ Thus, the solution is: $$x = \frac{26 - 5y}{2}, \quad y = y$$ ### Final Answer: The system is **dependent**, and the solution can be expressed as: $$x = \frac{26 - 5y}{2}, \quad y = y$$ where $$y$$ is any real number. --- Would you like further details on this solution? Here are some related questions to consider: 1. What is the difference between dependent and independent systems of equations? 2. How do you solve a system of equations by substitution instead of elimination? 3. Can this method be used for systems with three variables? 4. How can you check if a system is inconsistent? 5. How do you express the solution if the system had been inconsistent? **Tip**: Always check if one equation is a multiple of the other before proceeding with elimination, as it may simplify your work.

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Math Problem Analysis

Mathematical Concepts

Algebra
Systems of Equations
Method of Elimination
Dependent and Inconsistent Systems

Formulas

Elimination method: ax + by = c

Theorems

Dependent system theorem: when two equations are multiples of each other, they represent the same line and have infinitely many solutions

Suitable Grade Level

Grades 9-12

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