Solve the following system of equations and give a geometrical interpretation of the result.
3x + y - z = 2
x - 2y + z = 0
x + 2y - z = 2
so I solved it and got (1,2,3), but I don't understand what they mean by give a geometrical interpretation of the result.
Each equation represents a plane. If there is a unique solution, they all intersect in a point.
To solve the system of equations, we can use either substitution or elimination method. Let's use the elimination method in this case.
Step 1: Multiply the second equation by 3 and add it to the first equation to eliminate x:
3(3x + y - z) + (x - 2y + z) = 3(2) + 0
9x + 3y - 3z + x - 2y + z = 6
10x + y - 2z = 6 --> Equation (4)
Step 2: Multiply the third equation by 3 and add it to the first equation to eliminate x:
3(3x + y - z) + (x + 2y - z) = 3(2) + 2
9x + 3y - 3z + x + 2y - z = 6 + 2
10x + 5y - 4z = 8 --> Equation (5)
Step 3: Multiply Equation (4) by 5 and Equation (5) by 2, then subtract Equation (5) from Equation (4) to eliminate y:
5(10x + y - 2z) - 2(10x + 5y - 4z) = 5(6) - 2(8)
50x + 5y - 10z - 20x - 10y + 8z = 30 - 16
30x - 2z = 14 --> Equation (6)
Step 4: Solve Equations (4) and (6) simultaneously:
10x + y - 2z = 6 --> Equation (4)
30x - 2z = 14 --> Equation (6)
Multiplying Equation (4) by 3 and subtracting Equation (6) from it:
3(10x + y - 2z) - (30x - 2z) = 3(6) - 14
30x + 3y - 6z - 30x + 2z = 18 - 14
3y - 4z = 4 --> Equation (7)
Step 5: Solve Equations (6) and (7) simultaneously:
30x - 2z = 14 --> Equation (6)
3y - 4z = 4 --> Equation (7)
Multiplying Equation (6) by 3 and adding Equation (7) to it:
3(30x - 2z) + (3y - 4z) = 3(14) + 4
90x - 6z + 3y - 4z = 42 + 4
90x + 3y - 10z = 46 --> Equation (8)
Step 6: Divide Equation (8) by 3:
30x + y - 3z = 46/3
Now we have the system of equations:
30x + y - 3z = 46/3 --> Equation (9)
3y - 4z = 4 --> Equation (7)
Multiplying Equation (9) by 3 and adding Equation (7) to it:
3(30x + y - 3z) + (3y - 4z) = 3(46/3) + 4
90x + 3y - 9z + 3y - 4z = 46 + 4
90x + 6y - 13z = 58 --> Equation (10)
Step 7: Divide Equation (10) by 2:
45x + 3y - 6.5z = 29
So, the solution to the system of equations is (x, y, z) = (1, 2, 3).
Now, let's interpret the solution geometrically.
The system of equations corresponds to three planes in three-dimensional space. Each equation represents a plane, and the solution (1, 2, 3) is the point of intersection of these planes.
The geometric interpretation is that the three planes represented by the equations intersect at a single point, which is (1, 2, 3) in this case. This means that the system of equations has a unique solution, and the three planes intersect at a common point in three-dimensional space.
To solve the given system of equations, you have obtained the solution (1, 2, 3). This means that the values of x, y, and z that satisfy all three equations simultaneously are x = 1, y = 2, and z = 3.
Now, let's discuss the geometrical interpretation of this result. In a system of linear equations, each equation represents a plane in three-dimensional space. When multiple planes intersect at a unique point, that point represents the solution to the system.
In this case, the given system of equations represents three planes in three-dimensional space. The solution (1, 2, 3) indicates the coordinates of the point where all three planes intersect. Therefore, geometrically, the solution represents the point of intersection of these three planes.
Since we are dealing with three planes, it might be a bit challenging to visualize. However, you can imagine these planes as solid sheets in space. The point (1, 2, 3) represents the common point where all three sheets meet.
Note that if the system had no solution or infinitely many solutions, the geometrical interpretation would be different. In these cases, there would be no point of intersection or an entire line/plane of intersection, respectively.
So, in summary, the geometrical interpretation of the solution (1, 2, 3) is that it represents the point of intersection of the three planes represented by the given system of equations.