Math Problem Statement
Solve the system of linear equations given by the augmented matrix [2 -7 | 5; 6 3 | 1].
Solution
The problem presented in the image requires solving a system of linear equations using the given augmented matrix. Here is the step-by-step process to solve the system:
Augmented Matrix
The augmented matrix given is:
2 & -7 & 0 & 5 \\ 6 & 3 & 2 & 1 \\ \end{array}\right]$$ This corresponds to the following system of linear equations: $$\begin{aligned} 2x_1 - 7x_2 + 0x_3 &= 5 \quad \text{(Equation 1)} \\ 6x_1 + 3x_2 + 2x_3 &= 1 \quad \text{(Equation 2)} \end{aligned}$$ ### Step 1: Simplify Equation 1 From Equation 1: $$2x_1 - 7x_2 = 5$$ We can solve for $$x_1$$: $$x_1 = \frac{5 + 7x_2}{2}$$ ### Step 2: Substitute $$x_1$$ into Equation 2 Substitute $$x_1$$ into the second equation: $$6\left(\frac{5 + 7x_2}{2}\right) + 3x_2 + 2x_3 = 1$$ Simplify: $$3(5 + 7x_2) + 3x_2 + 2x_3 = 1$$ $$15 + 21x_2 + 3x_2 + 2x_3 = 1$$ $$24x_2 + 2x_3 = 1 - 15$$ $$24x_2 + 2x_3 = -14$$ ### Step 3: Further Simplify Divide the entire equation by 2: $$12x_2 + x_3 = -7$$ ### Solution: The system now looks like this: $$\begin{aligned} x_1 &= \frac{5 + 7x_2}{2}, \\ 12x_2 + x_3 &= -7. \end{aligned}$$ This system has $$x_2$$ and $$x_3$$ as free variables, meaning the system has infinitely many solutions. ### Parametric Form of the Solution: Let $$x_2 = t$$ (where $$t$$ is a free parameter), then: $$x_1 = \frac{5 + 7t}{2},$$ $$x_3 = -7 - 12t.$$ So the solution can be expressed as: $$(x_1, x_2, x_3) = \left(\frac{5 + 7t}{2}, t, -7 - 12t\right), \quad \text{where } t \in \mathbb{R}.$$ ### Final Answer: The solution set is: $$\left(\frac{5 + 7t}{2}, t, -7 - 12t\right), \quad \text{where } t \in \mathbb{R}.$$ --- Would you like any more details on this solution, or do you have any questions? Here are five related questions for further exploration: 1. How can we determine if a system of linear equations has a unique solution, no solution, or infinitely many solutions? 2. What is the significance of the rank of a matrix in solving systems of equations? 3. How can the method of Gaussian elimination be applied to solve a system of linear equations? 4. What are the conditions under which a matrix is invertible, and how does that relate to solving linear systems? 5. How do we interpret the solution set geometrically when the system has free variables? **Tip:** When dealing with linear systems, always consider simplifying the matrix using row operations before attempting to solve the equations directly. This can save time and reduce errors.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Systems of Linear Equations
Augmented Matrix
Formulas
x1 = (5 + 7x2)/2
12x2 + x3 = -7
Theorems
Solution of Linear Systems
Suitable Grade Level
Undergraduate Level
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