Math Problem Statement

Solve the system of equations. If the system has no​ solution, say that it is inconsistent. left brace Start 2 By 1 Matrix 1st Row 1st Column 2nd Row 1st Column EndMatrix

x plus 2 y equals 6 3 x plus 6 y equals 18 Question content area bottom Part 1 Select the correct choice below​ and, if​ necessary, fill in any answer boxes within your choice. A. The solution is xequals

enter your response here and yequals

enter your response here. ​(Type integers or simplified​ fractions.) B. There are infinitely many solutions. Using ordered​ pairs, they can be expressed as StartSet left parenthesis x comma y right parenthesis vertical line x equals nothing comma y any real number EndSet

. ​(Simplify your answer. Type an expression using y as the variable as​ needed.) C. The system is inconsistent.

Solution

We are given the following system of equations:

1. & \quad x + 2y = 6 \\ 2. & \quad 3x + 6y = 18 \end{aligned}$$ ### Step 1: Analyze the system We will first check whether the two equations are independent or dependent, and whether a solution exists. #### Multiply the first equation by 3: Multiplying the entire first equation by 3 to match the form of the second equation: $$3(x + 2y) = 3 \times 6$$ $$3x + 6y = 18$$ This gives us the equation $$3x + 6y = 18$$, which is identical to the second equation in the system. ### Step 2: Conclusion Since the second equation is just a multiple of the first equation, the two equations are dependent. This means that they represent the same line, and there are infinitely many solutions. The solution can be written in terms of one of the variables. Solving the first equation for $$x$$: $$x = 6 - 2y$$ Thus, the solution is expressed as an infinite set of ordered pairs: $$\boxed{B: \text{There are infinitely many solutions. Using ordered pairs, they can be expressed as } \{(x, y) \mid x = 6 - 2y, y \text{ any real number} \}.}$$ Would you like further details on how to approach this, or do you have any questions? Here are 5 related questions: 1. How do you determine if a system of equations is dependent or independent? 2. What methods can be used to solve systems of linear equations? 3. What does it mean geometrically when two equations are dependent? 4. How do you check if a system of equations has no solution? 5. Can you provide an example of an inconsistent system of equations? **Tip:** If two equations in a system are multiples of each other, the system will have infinitely many solutions.

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Math Problem Analysis

Mathematical Concepts

Linear Equations
Systems of Equations
Dependent and Independent Systems

Formulas

x + 2y = 6
3x + 6y = 18

Theorems

Consistency of Systems
Dependency of Linear Equations

Suitable Grade Level

Grades 9-11

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