Math Problem Statement
Solve the system of equations: -3x + 14y = -28 and -2(x - 2y) + 3y = -14
Solution
We are asked to solve the system of equations given by:
-3x + 14y = -28 \\ -2(x - 2y) + 3y = -14 \end{cases}$$ ### Step 1: Simplify the second equation We start by simplifying the second equation. $$-2(x - 2y) + 3y = -14$$ Distribute the $$-2$$: $$-2x + 4y + 3y = -14$$ Combine like terms: $$-2x + 7y = -14$$ ### Step 2: Solve using the substitution or elimination method Now, we have the following system: $$\begin{cases} -3x + 14y = -28 \\ -2x + 7y = -14 \end{cases}$$ We can use the elimination method. Let's multiply the second equation by 2 to align the coefficients of $$y$$: $$2(-2x + 7y = -14) \Rightarrow -4x + 14y = -28$$ Now, subtract the first equation from this result: $$(-4x + 14y) - (-3x + 14y) = (-28) - (-28)$$ Simplify: $$-4x + 3x = 0 \quad \text{and} \quad -28 + 28 = 0$$ This gives: $$-x = 0 \quad \Rightarrow \quad x = 0$$ ### Step 3: Solve for $$y$$ Substitute $$x = 0$$ into one of the original equations, for example, the second one: $$-2(0) + 7y = -14 \quad \Rightarrow \quad 7y = -14 \quad \Rightarrow \quad y = -2$$ ### Solution: The solution to the system is $$x = 0$$ and $$y = -2$$, or $$(0, -2)$$. Thus, the correct answer is: $$\boxed{A) \ (0, -2)}$$ --- Would you like any further details or explanations on this? Here are 5 related questions for further practice: 1. How can we solve a system of equations using substitution instead of elimination? 2. What are the conditions for a system of equations to have no solution or infinite solutions? 3. How can we check the solution of a system of equations graphically? 4. Can you explain how to use matrices to solve systems of equations? 5. How do linear transformations relate to systems of linear equations? **Tip:** When solving systems of equations, always check if simplifying one equation makes the system easier to solve.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Equations
Systems of Equations
Elimination Method
Substitution Method
Formulas
Linear equation form: ax + by = c
Theorems
Elimination Method
Substitution Method
Suitable Grade Level
Grades 8-10
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