Math Problem Statement

Solve the system with the addition method: -9x - 7y = 57 -4x + 5y = 31

Solution

The problem in the image asks to solve the system of linear equations using the addition method:

-9x - 7y &= 57 \\ -4x + 5y &= 31 \end{aligned}$$ ### Steps to Solve Using the Addition Method: 1. **Eliminate one variable**: We want to eliminate either $$x$$ or $$y$$. Let's eliminate $$x$$ by making the coefficients of $$x$$ in both equations equal. The least common multiple of $$-9$$ and $$-4$$ is 36, so we multiply the first equation by 4 and the second equation by 9: $$4(-9x - 7y) = 4(57) \implies -36x - 28y = 228$$ $$9(-4x + 5y) = 9(31) \implies -36x + 45y = 279$$ 2. **Add the equations** to eliminate $$x$$: $$(-36x - 28y) + (-36x + 45y) = 228 + 279$$ Simplifying: $$-36x + 36x + (-28y + 45y) = 507 \implies 17y = 507$$ 3. **Solve for $$y$$**: $$y = \frac{507}{17} = 29.8$$ So $$y = 29.8$$. 4. **Substitute $$y = 29.8$$** back into one of the original equations to find $$x$$. Let's use the second equation: $$-4x + 5(29.8) = 31$$ Simplifying: $$-4x + 149 = 31$$ $$-4x = 31 - 149 \implies -4x = -118$$ $$x = \frac{-118}{-4} = 29.5$$ Thus, $$x = 29.5$$ and $$y = 29.8$$. ### Final Solution: The system has one solution, which is $$x = 29.5$$ and $$y = 29.8$$. Would you like further clarification on any step? Here are 5 related questions to practice: 1. Can you solve a similar system using substitution instead of addition? 2. How do we handle a system where the variables cancel out completely? 3. What are the differences between the elimination method and substitution method? 4. How do you verify the solution to a system of equations? 5. What happens if the two equations represent parallel lines? **Tip:** Always check your final solution by plugging both values of $$x$$ and $$y$$ into the original equations to ensure consistency.

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Math Problem Analysis

Mathematical Concepts

Algebra
Systems of Linear Equations
Addition (Elimination) Method

Formulas

Ax + By = C form for linear equations

Theorems

System of linear equations solution methods (addition/elimination)

Suitable Grade Level

Grades 8-10

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