Math Problem Statement
Solve the system below using augmented matrix methods. 4x1-3x2=-7 -12x1+9x2=21
Solution
To solve the system of equations using the augmented matrix method, follow these steps:
System of Equations:
Step 1: Write the Augmented Matrix
The augmented matrix for the system is:
4 & -3 & | & -7 \\ -12 & 9 & | & 21 \end{pmatrix}$$ ### Step 2: Row Operations We aim to make this matrix into reduced row-echelon form (RREF). #### Step 2.1: Normalize the first row Let's normalize the first row by dividing it by 4 (to make the leading entry of the first row 1): $$R_1 \to \frac{R_1}{4}$$ $$\begin{pmatrix} 1 & -\frac{3}{4} & | & -\frac{7}{4} \\ -12 & 9 & | & 21 \end{pmatrix}$$ #### Step 2.2: Eliminate the $$x_1$$-term from the second row Now, to eliminate the $$x_1$$-term from the second row, perform the row operation: $$R_2 \to R_2 + 12R_1$$ $$\begin{pmatrix} 1 & -\frac{3}{4} & | & -\frac{7}{4} \\ 0 & 0 & | & 0 \end{pmatrix}$$ ### Step 3: Interpret the Result The second row becomes all zeros, which means the system is dependent. This implies there are infinitely many solutions. The system can be written as: $$x_1 - \frac{3}{4}x_2 = -\frac{7}{4}$$ ### Step 4: Express $$x_1$$ in terms of $$x_2$$ Solve for $$x_1$$: $$x_1 = \frac{3}{4}x_2 - \frac{7}{4}$$ Thus, the solution is given by the parametric form: $$x_1 = \frac{3}{4}t - \frac{7}{4}, \quad x_2 = t$$ where $$t$$ is any real number. ### Conclusion: The system has infinitely many solutions and can be expressed parametrically as: $$(x_1, x_2) = \left( \frac{3}{4}t - \frac{7}{4}, t \right) \quad \text{for any } t \in \mathbb{R}.$$ --- Do you have any questions or would you like further details? Here are 5 related questions to expand your understanding: 1. What is the general condition for a system of equations to have infinitely many solutions? 2. How can you verify the solution to this system using substitution? 3. How does the rank of the matrix affect the number of solutions in a system of equations? 4. Can you describe how the augmented matrix method is different from Gaussian elimination? 5. How would the solution change if the second equation were slightly different? **Tip:** When a system leads to a row of zeros, it usually indicates that the system is either dependent or inconsistent, depending on the augmented part of the matrix.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Systems of Equations
Augmented Matrix Method
Formulas
Row Reduction
Parametric Form of Solutions
Theorems
Dependent System Theorem
Row Echelon Form
Suitable Grade Level
Grades 9-12
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