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.


### Part a) System of equations

We are given the following two planes:
1. \( 5x + 2y + z = -1 \)
2. \( 2x - y + 6z = 7 \)

We can form a system of equations from this:
\[
\begin{aligned}
5x + 2y + z &= -1 \\
2x - y + 6z &= 7 
\end{aligned}
\]

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).

Consider the following planes.
5x + 2y + z = −1 and 2x − y + 6z = 7
a) Use these equations to 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.

Learn how to find the line of intersection between two planes defined by the equations 5x + 2y + z = -1 and 2x - y + 6z = 7. This solution involves reducing the augmented matrix to row echelon form, identifying free variables, and expressing the solution in parametric form. Suitable for undergraduate-level students studying linear algebra.

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