Math Problem Statement
Solve the system of linear equations: 3x - y = 2 and x + 2y = 4.
Solution
The image shows two systems of linear equations, labeled as options C and D:
C:
3x - y &= 2 \\ x + 2y &= 4 \end{aligned}$$ **D:** $$\begin{aligned} 3x + y &= -2 \\ x + 2y &= 4 \end{aligned}$$ To solve each system of equations, we can use either substitution or elimination methods. I'll solve both systems and compare the results. ### Option C: The system is: $$\begin{aligned} 3x - y &= 2 \quad (1) \\ x + 2y &= 4 \quad (2) \end{aligned}$$ 1. Solve equation (1) for $$y$$: $$y = 3x - 2$$ 2. Substitute $$y = 3x - 2$$ into equation (2): $$x + 2(3x - 2) = 4$$ $$x + 6x - 4 = 4$$ $$7x - 4 = 4$$ $$7x = 8$$ $$x = \frac{8}{7}$$ 3. Substitute $$x = \frac{8}{7}$$ into $$y = 3x - 2$$: $$y = 3\left(\frac{8}{7}\right) - 2 = \frac{24}{7} - 2 = \frac{24}{7} - \frac{14}{7} = \frac{10}{7}$$ So, the solution for option C is: $$x = \frac{8}{7}, \quad y = \frac{10}{7}$$ ### Option D: The system is: $$\begin{aligned} 3x + y &= -2 \quad (1) \\ x + 2y &= 4 \quad (2) \end{aligned}$$ 1. Solve equation (1) for $$y$$: $$y = -3x - 2$$ 2. Substitute $$y = -3x - 2$$ into equation (2): $$x + 2(-3x - 2) = 4$$ $$x - 6x - 4 = 4$$ $$-5x - 4 = 4$$ $$-5x = 8$$ $$x = -\frac{8}{5}$$ 3. Substitute $$x = -\frac{8}{5}$$ into $$y = -3x - 2$$: $$y = -3\left(-\frac{8}{5}\right) - 2 = \frac{24}{5} - 2 = \frac{24}{5} - \frac{10}{5} = \frac{14}{5}$$ So, the solution for option D is: $$x = -\frac{8}{5}, \quad y = \frac{14}{5}$$ ### Summary: - **Option C** gives the solution $$x = \frac{8}{7}, y = \frac{10}{7}$$ - **Option D** gives the solution $$x = -\frac{8}{5}, y = \frac{14}{5}$$ Let me know if you'd like further explanation or have any other questions! Here are five related questions to expand on the topic: 1. What is the difference between the substitution and elimination methods in solving systems of equations? 2. How can you verify the solutions of a system of equations? 3. Are there any graphical methods to solve systems of linear equations? 4. How would you solve a system of equations with three variables instead of two? 5. What does it mean if a system of equations has no solution or infinite solutions? **Tip:** When solving systems of equations, always double-check your substitution or elimination steps to avoid small algebraic errors that can lead to incorrect answers!Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Algebra
Linear Equations
Systems of Equations
Formulas
Substitution method: Solve one equation for one variable and substitute into the other equation
Elimination method: Multiply or subtract equations to eliminate one variable
Theorems
Properties of linear equations
Solution methods for systems of linear equations
Suitable Grade Level
Grades 8-10
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