Math Problem Statement

Which of these strategies would eliminate a variable in the system of equations? 1) x - 2y = 11 2) 5x + 3y = -11 A) Multiply the top equation by 5, then add the equations. B) Add the equations. C) Multiply the top equation by -5, then add the equations.

Solution

We are given the following system of equations:

1. & \quad x - 2y = 11 \\ 2. & \quad 5x + 3y = -11 \end{aligned}$$ The goal is to determine which strategy would eliminate one of the variables by combining the equations. ### Strategy A: Multiply the top equation by 5, then add the equations. Multiplying the first equation by 5 gives: $$5(x - 2y) = 5 \cdot 11 \implies 5x - 10y = 55$$ Now we add the two equations: $$(5x - 10y) + (5x + 3y) = 55 + (-11)$$ $$10x - 7y = 44$$ This does not eliminate any variable, so **Strategy A** does not work. ### Strategy B: Add the equations. If we directly add the equations without any modification: $$(x - 2y) + (5x + 3y) = 11 + (-11)$$ $$6x + y = 0$$ This also does not eliminate any variable, so **Strategy B** does not work either. ### Strategy C: Multiply the top equation by -5, then add the equations. Multiplying the first equation by -5 gives: $$-5(x - 2y) = -5 \cdot 11 \implies -5x + 10y = -55$$ Now we add the two equations: $$(-5x + 10y) + (5x + 3y) = -55 + (-11)$$ $$0x + 13y = -66$$ This eliminates the $$x$$-variable, leaving us with a simple equation to solve for $$y$$. Thus, **Strategy C** is the correct choice, as it eliminates the $$x$$-variable. Would you like more details or have any questions? Here are 5 follow-up questions to consider: 1. How do you solve the resulting equation $$13y = -66$$? 2. How would you solve for $$x$$ once you find $$y$$? 3. Could this system be solved using substitution instead of elimination? 4. How can you verify the solution to the system of equations? 5. What is the significance of elimination versus substitution methods in solving systems of equations? **Tip**: Always ensure you multiply equations correctly when using elimination, so the variable you want to cancel out has opposite coefficients.

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Math Problem Analysis

Mathematical Concepts

Algebra
Systems of Equations
Elimination Method

Formulas

Elimination method: ax + by = c, dx + ey = f
Multiplication and addition of linear equations

Theorems

Elimination theorem for systems of equations

Suitable Grade Level

Grades 9-10

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