Math Problem Statement
2x - 3y = 7 and 2y - 3x = -8
Solution
You provided the system of equations from the image:
1) & \ \ 2x - 3y = 7 \\ 2) & \ \ 2y - 3x = -8 \end{aligned}$$ I will solve this system of equations using substitution or elimination method. ### Step 1: Rewriting the equations: Let's rewrite the second equation for clarity: $$-3x + 2y = -8$$ Now we have the system: $$\begin{aligned} 1) & \ \ 2x - 3y = 7 \\ 2) & \ \ -3x + 2y = -8 \end{aligned}$$ ### Step 2: Multiply both equations to align coefficients We can use the elimination method by multiplying both equations to make the coefficients of $$x$$ or $$y$$ equal. To eliminate $$x$$, multiply the first equation by 3 and the second by 2: $$3(2x - 3y) = 3(7) \quad \Rightarrow \quad 6x - 9y = 21$$ $$2(-3x + 2y) = 2(-8) \quad \Rightarrow \quad -6x + 4y = -16$$ ### Step 3: Add the two equations to eliminate $$x$$: $$(6x - 9y) + (-6x + 4y) = 21 + (-16)$$ $$-5y = 5$$ ### Step 4: Solve for $$y$$: $$y = \frac{5}{-5} = -1$$ ### Step 5: Substitute $$y = -1$$ into one of the original equations: Substitute into the first equation: $$2x - 3(-1) = 7$$ $$2x + 3 = 7$$ $$2x = 7 - 3 = 4$$ $$x = \frac{4}{2} = 2$$ ### Final Answer: The solution to the system of equations is: $$x = 2, \quad y = -1$$ Would you like further details or have any questions? Here are 5 related questions for practice: 1. Can you solve a similar system of equations with different coefficients? 2. What happens if the system of equations has no solution? 3. How would you solve a system using the substitution method? 4. Can you solve a system where both variables are in terms of fractions? 5. How would you solve this system graphically? **Tip**: Always verify your solution by substituting the values of $$x$$ and $$y$$ back into the original equations!Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Algebra
Systems of Equations
Elimination Method
Formulas
Linear equation in the form of ax + by = c
Theorems
None specifically, but the elimination method is a common algebraic technique
Suitable Grade Level
Grades 8-10