Math Problem Statement
Solve the system of linear equations: 3x + 9y = 3 and 9x + 6y = 2
Solution
The system of linear equations provided in the image is:
3x + 9y = 3 \\ 9x + 6y = 2 \end{cases}$$ Let's solve this system using the method of elimination or substitution. ### Step 1: Simplify the Equations First, simplify each equation if possible: 1. Divide the first equation by 3: $$x + 3y = 1$$ 2. Divide the second equation by 3: $$3x + 2y = \frac{2}{3}$$ Now, the simplified system is: $$\begin{cases} x + 3y = 1 \\ 3x + 2y = \frac{2}{3} \end{cases}$$ ### Step 2: Use Substitution or Elimination Let's solve this using substitution. From the first equation: $$x = 1 - 3y$$ Substitute $$x = 1 - 3y$$ into the second equation: $$3(1 - 3y) + 2y = \frac{2}{3}$$ Simplify this: $$3 - 9y + 2y = \frac{2}{3}$$ $$-7y = \frac{2}{3} - 3$$ $$-7y = \frac{2}{3} - \frac{9}{3}$$ $$-7y = -\frac{7}{3}$$ $$y = \frac{1}{3}$$ ### Step 3: Find $$x$$ Now substitute $$y = \frac{1}{3}$$ into $$x = 1 - 3y$$: $$x = 1 - 3\left(\frac{1}{3}\right)$$ $$x = 1 - 1 = 0$$ ### Solution The solution to the system is: $$x = 0, \quad y = \frac{1}{3}$$ Would you like a detailed breakdown of any specific step? Here are some related questions: 1. How can we verify the solution by plugging values back into the original equations? 2. What are other methods to solve this system, such as using matrices? 3. Can this system be solved graphically? What does the solution represent on a graph? 4. What happens if the system has no solution or infinitely many solutions? 5. How can we interpret these equations in a real-world context? **Tip:** Always simplify equations when possible—it makes the calculations easier and less prone to errors!Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear equations
Systems of equations
Substitution method
Elimination method
Formulas
Substitution: Solve for one variable in terms of another and substitute in the second equation
Elimination: Multiply equations to eliminate one variable and solve for the other
Theorems
System of linear equations theorem
Suitable Grade Level
Grades 8-10
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