Math Problem Statement

Use the method of substitution 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. 4x + 2y = 12, 4x + 4y = 8

Solution

The system of equations provided is:

4x + 2y = 12 \\ 4x + 4y = 8 \end{cases}$$ ### Step-by-step solution using substitution: 1. **Simplify the second equation**: The second equation $$4x + 4y = 8$$ can be simplified by dividing the entire equation by 4: $$x + y = 2$$ 2. **Solve for $$x$$ or $$y$$**: From $$x + y = 2$$, solve for $$x$$: $$x = 2 - y$$ 3. **Substitute into the first equation**: Substitute $$x = 2 - y$$ into the first equation $$4x + 2y = 12$$: $$4(2 - y) + 2y = 12$$ Simplify the equation: $$8 - 4y + 2y = 12$$ Combine like terms: $$8 - 2y = 12$$ 4. **Solve for $$y$$**: Subtract 8 from both sides: $$-2y = 4$$ Divide by $$-2$$: $$y = -2$$ 5. **Substitute $$y = -2$$ into $$x = 2 - y$$**: $$x = 2 - (-2) = 2 + 2 = 4$$ ### Conclusion: The solution to the system of equations is $$x = 4$$ and $$y = -2$$, so the system has **only one solution**. Would you like more details or have any other questions? --- Here are 5 related questions to explore: 1. How can you check if a system of equations has no solution? 2. What is the difference between a dependent and an inconsistent system? 3. Can a system of linear equations have more than one solution? 4. How does the substitution method compare to the elimination method? 5. How can graphing help in solving a system of linear equations? **Tip**: Always simplify equations if possible before using substitution to make the process easier.

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Math Problem Analysis

Mathematical Concepts

Substitution method
Solving systems of equations
Linear equations

Formulas

Substitution method formula

Theorems

No specific theorems but general properties of linear systems

Suitable Grade Level

Grades 9-11

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