Math Problem Statement
Consider the following planes. 5x + 2y + z = −1 and 2x − y + 6z = 7 a) Use these equations for form a system. Reduce the corresponding augmented matrix to row echelon form. (Order the columns from x to z.)
Identify the free variables from the row reduced matrix. (Select all that apply.) x y z
Use back substitution to write the variables corresponding to the leading variables in terms of the free variables. State the resulting solution.
[x ,y, z]=[13/9,- 37/9, 0]+ t[,,_]
What does this solution represent?
The skew line of the given planes.
The two lines of intersection of the given planes.
The line of intersection of the given planes.
The plane perpendicular to the given planes.
The single point of intersection of the given planes.
Solution
Part a) System of equations
We are given the following two planes:
We can form a system of equations from this:
5x + 2y + z &= -1 \\ 2x - y + 6z &= 7 \end{aligned}$$ Now, we can express this system as an augmented matrix, where each equation is written as a row: $$\begin{bmatrix} 5 & 2 & 1 & | & -1 \\ 2 & -1 & 6 & | & 7 \end{bmatrix}$$ ### Step 1: Row Reduction to Row Echelon Form We will now reduce this matrix to row echelon form using Gaussian elimination. 1. Multiply the first row by $$\frac{2}{5}$$ and subtract it from the second row to eliminate the $$x$$-term in the second row. Let's perform the row operation: $$R_2 = R_2 - \left( \frac{2}{5} \right) R_1$$ The updated matrix is: $$\begin{bmatrix} 5 & 2 & 1 & | & -1 \\ 0 & -\frac{9}{5} & \frac{28}{5} & | & \frac{39}{5} \end{bmatrix}$$ 2. To simplify the second row, multiply by $$-\frac{5}{9}$$ to get the leading coefficient to 1: $$R_2 = \left( -\frac{5}{9} \right) R_2$$ The resulting matrix is: $$\begin{bmatrix} 5 & 2 & 1 & | & -1 \\ 0 & 1 & -\frac{28}{9} & | & -\frac{13}{3} \end{bmatrix}$$ 3. Subtract $$2R_2$$ from $$R_1$$ to eliminate the $$y$$-term in the first row: $$R_1 = R_1 - 2R_2$$ The updated matrix is: $$\begin{bmatrix} 5 & 0 & \frac{19}{9} & | & -\frac{37}{9} \\ 0 & 1 & -\frac{28}{9} & | & -\frac{13}{3} \end{bmatrix}$$ Finally, divide $$R_1$$ by 5 to make the leading coefficient of $$x$$ equal to 1: $$R_1 = \frac{1}{5}R_1$$ The row echelon form of the augmented matrix is: $$\begin{bmatrix} 1 & 0 & \frac{19}{45} & | & -\frac{37}{45} \\ 0 & 1 & -\frac{28}{9} & | & -\frac{13}{3} \end{bmatrix}$$ ### Identifying Free Variables In the row-reduced matrix, we have leading variables $$x$$ and $$y$$, while $$z$$ does not have a leading entry (it's free). Therefore: **Free variable: $$z$$** ### Back Substitution to Write the Solution From the row-reduced matrix: 1. The first row gives: $$x + \frac{19}{45}z = -\frac{37}{45} \quad \Rightarrow \quad x = -\frac{37}{45} - \frac{19}{45}z$$ Simplifying: $$x = -\frac{37}{45} - \frac{19}{45}z$$ 2. The second row gives: $$y - \frac{28}{9}z = -\frac{13}{3} \quad \Rightarrow \quad y = -\frac{13}{3} + \frac{28}{9}z$$ Thus, the solution in terms of $$z = t$$ (a free parameter) is: \[ [x, y, z] = \left[ -\frac{37}{45}, -\frac{13}{3}, 0 \right] + t \left[ -\frac{19}{45}, \frac{28}{9}, 1 \right] \] ### Interpretation of the Solution This is the **parametric equation of the line** that represents the **line of intersection of the given planes**. Therefore, the correct interpretation is: **The line of intersection of the given planes.** --- Would you like to ask more details or clarify any step? Here are five related questions for further exploration: 1. How do you identify free variables in any system of linear equations? 2. What is the geometric meaning of a line of intersection of two planes? 3. Can a system of equations with two planes have more than one solution type? 4. How would the solution differ if the system represented parallel planes? 5. How can the row reduction method be applied to solve systems with more variables? **Tip**: When solving systems of equations geometrically, parallel planes will either have no solution (if distinct) or infinitely many (if coincident).Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Systems of Equations
Gaussian Elimination
Row Echelon Form
Parametric Equations
Formulas
Row Reduction for solving linear systems
Parametric equation of a line
Theorems
Fundamental Theorem of Linear Algebra
Suitable Grade Level
Undergraduate
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Line of Intersection Between Two Planes -4x + 5y - z = 0 and -5x - 5y + 5z = 5