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

Question

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

Accepted Answer

∻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.⚹◊What is the geometric interpretation of the solution set of the system of equations representing the intersection of three planes in three-dimensional space?⍭ Generate me a similar question◊

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