Math Problem Statement
The system of equations 2ax + by = 9 and bx - 3ay = 4 has the solution x = 3 and y = -2. Determine a and b.
Solution
The system of equations provided in the image is:
2ax + by = 9 \\ bx - 3ay = 4 \end{cases}$$ It also says that the solution to this system is $$x = 3$$ and $$y = -2$$. The task is to determine the values of $$a$$ and $$b$$. ### Step 1: Substitute $$x = 3$$ and $$y = -2$$ into both equations. #### First equation: $$2a(3) + b(-2) = 9$$ This simplifies to: $$6a - 2b = 9 \quad \text{(Equation 1)}$$ #### Second equation: $$b(3) - 3a(-2) = 4$$ This simplifies to: $$3b + 6a = 4 \quad \text{(Equation 2)}$$ ### Step 2: Solve the system of equations We now have two equations: 1. $$6a - 2b = 9$$ 2. $$6a + 3b = 4$$ #### Subtract the first equation from the second: $$(6a + 3b) - (6a - 2b) = 4 - 9$$ This simplifies to: $$5b = -5$$ $$b = -1$$ #### Substitute $$b = -1$$ into one of the original equations, for example, the first equation: $$6a - 2(-1) = 9$$ $$6a + 2 = 9$$ $$6a = 7$$ $$a = \frac{7}{6}$$ ### Final Answer: $$a = \frac{7}{6}, \quad b = -1$$ Let me know if you'd like further details or clarification! ### Related Questions: 1. How would the solution change if the value of $$x$$ or $$y$$ were different? 2. What method can you use to solve a system of equations when substitution is not possible? 3. How do you verify that your solution to the system of equations is correct? 4. What role do the coefficients of $$a$$ and $$b$$ play in determining the solution to the system? 5. How would the problem change if there were more unknowns in the system of equations? **Tip:** Always double-check your algebraic manipulations when solving systems of equations to avoid small calculation errors.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Systems of Equations
Linear Algebra
Substitution Method
Formulas
Equation 1: 6a - 2b = 9
Equation 2: 3b + 6a = 4
Theorems
Substitution and Elimination in Systems of Equations
Suitable Grade Level
Grades 9-12