Question

Do the three planes x 1 plus 4 x 2 plus 3 x 3 equals 5x1+4x2+3x3=5​, x 2 minus 3 x 3 equals 1x2−3x3=1​, and 3 x 1 plus 15 x 2 equals 123x1+15x2=12 have at least one common point of​ intersection? Explain.

Answer 1

The system of equations formed by the three planes does not have a common solution, so the planes do not have a common point of intersection.

Step-by-step explanation:

To determine if the three planes have at least one common point of intersection, we need to solve the system of equations formed by the planes. The system of equations is:

x1 + 4x2 + 3x3 = 5

x2 - 3x3 = 1

3x1 + 15x2 = 12

We can solve this system using various methods such as substitution or elimination. Let's use the elimination method:

Multiply the second equation by 4 to make the coefficients of x1 and x2 the same as in the third equation. The system becomes:

x1 + 4x2 + 3x3 = 5

4x2 - 12x3 = 4

3x1 + 15x2 = 12

Now subtract the second equation from the first equation to eliminate x2. The system becomes:

x1 -_ 12x3 = 1

4x2 - 12x3 = 4

3x1 +__9x3 = 9

Now subtract the third equation multiplied by 3 from the second equation multiplied by 4 to eliminate x1. The system becomes:

x1 -_ 12x3 = 1

4x2 - 12x3 = 4

__9x3 - 9x3 = -15

Simplify the third equation to:

0 = -15

Since this equation is not true, it means that the system of equations does not have a solution. Therefore, the three planes do not have a common point of intersection.

Answer 2

nswer:

No, they do not have any common point of intersection.



Step-by-step explanation:

Do the three planes have a common point of intersection?

X1 + 4X2 + 3X3 = 5

X2 - 3X3 = 1

3X1 + 15X2 = 15

Using matrix method

First, convert the given equation to matrix equivalent

`\left[\begin{array}{cccc}1&4&3&5\\0&1&3&1\\3&15&0&15\end{array}\right]`

Subtract Row1 from Row 3

Row3 = Row3 - Row1

`\left[\begin{array}{cccc}1&4&3&5\\0&1&3&1\\2&11&-3&10\end{array}\right]`

Inconsistent planes.

Planes from row 2 and row 3 do not intersect and they do no have anything in common.

Hence, we do not need to move further