```
Question 9
Find the intersection of the three planes:
$\begin{array}{l}
x+y+2 z=-2 \\
3 x-y+14 z=6 \\
x+2 y=-5
\end{array}$
```

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Answer from Sia

Posted 7 months ago

Solution by Steps

step 1

To find the intersection of the three planes, we can solve the system of equations given by the plane equations

step 2

The system of equations is: $\begin{cases} x + y + 2z = -2 \\ 3x - y + 14z = 6 \\ x + 2y = -5 \end{cases}$

step 3

From the asksia-ll calculation list, we have the partial solution: $y = -\frac{x}{2} - \frac{5}{2}$ and $z = \frac{1}{4} - \frac{x}{4}$

step 4

Substitute $y$ and $z$ from step 3 into the first equation $x + y + 2z = -2$ to find the value of $x$

step 5

Solving for $x$ gives us $x = -3$

step 6

Substitute $x = -3$ back into the equations for $y$ and $z$ to find their values

step 7

We get $y = -\frac{-3}{2} - \frac{5}{2} = \frac{3}{2} - \frac{5}{2} = -1$ and $z = \frac{1}{4} - \frac{-3}{4} = \frac{1}{4} + \frac{3}{4} = 1$

Answer

The intersection of the three planes is the point $(-3, -1, 1)$.

Key Concept

Solving a system of linear equations

Explanation

To find the intersection of three planes, we solve the system of equations represented by the planes. The solution gives us the coordinates of the intersection point.

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