Question

Use the reduced row echelon form you found in the previous question to determine the solution(s) of the system of linear equations

4x−12y+35z = 24
12x−32y+95z = 65
8x−28y+85z = 57
If there is no solution, write none - If there is one solution, write x=a∧y=b∧z=c for appropriate numbers a,b, and c - If there are multiple solutions, then express the basic variables by use of the free parameters, with no equation expressing the free parameters. For instance x=3z∧y=z+2 (notice we don't write z=z in this case). - If all values of x,y, and z are solutions, write all Solve the following system of linear equations in the unknowns x,y, and z.
x+y−2z = 0
3x+4y−8z = 1
2x = -2
​- If there is no solution, write none - If there is one solution, write x=a∧y=b∧z=c for appropriate numbers a,b, and c - If there are multiple solutions, then express the basic variables by use of the free parameters, with no equation expressing the free parameters. For instance x=3z∧y=z+2 (notice we don't write z=z in this case). - If all values of x,y, and z are solutions, write all The plane V in R
3
is given by the following equation in the coordinates [x,y,z] : y−z+7=0. The line ℓ in R
3
is given by the following system of equations in the coordinates [x,y,z] : −x−y+4=0∧x−z−4=0. Determine the intersection of the line ℓ with the plane V. Give your answer in the form x=a∧y=b∧z=c or as [a,b,c] for appropriate rational numbers a,b, and c.

Answer 1

To find the point of intersection of line ℓ with plane V, we solve for the variables using the given equations. The intersection point is (2, -6, 1) or [2, -6, 1].

Step-by-step explanation:

To determine the intersection of line ℓ with plane V, we need to solve the system of linear equations presented by the line and the equation of the plane. The line ℓ is given by −x−y+4=0 and x−z−4=0, while the plane V is given by y−z+7=0. Solving these three equations together will yield the intersection point.

First, we rearrange the equation for the plane to solve for y, giving as y=z−7. Next, we substitute that into the first line equation to eliminate y: −x−(z−7)+4=0, which simplifies to x=3−z. Using the second line equation, we substitute x for 3−z: (3−z)−z−4=0, which simplifies to z=1.

Using the value of z, we can solve for x using the expression x=3−z to get x=2. Then we substitute z into the plane's equation to obtain the value for y: y=1−7, which implies y=−6. Therefore, the intersection of line ℓ with plane V is the point (2, −6, 1), which can also be represented as [2,−6,1].