Solving the system of linear equations but can't figure out this one.

Question

Solve the system of linear equations and check any solutions algebraically. (If there is no solution, enter NO SOLUTION. If the system is dependent, express x, y, and z in terms of the parameter a.)

x-2y+5z=25

4x -z=0

x,y,z=

This problem is causing a lot of confusion for me. I've tried to solve it but can only conclude that there is no solution as three variables require three equations to have a solution but "no solution" has been marked as incorrect in the homework assignment.

Answer 1

Here is one way to set this up:

Solve second equation for z: z = 4x

Substitute into the 1st equation:

x - 2y +5(4x)=25

21x-2y=25

Solve for y:

y = 1/2(21x-25)

Now we have:

x = α

y = 1/2(21α-25)

z=4α

Check a solution:

Let α=2

x=2

y=17/2

z=8

2 - 2(17/2) + 5(8) =? 25

-15+40 =? 25 (check)

4(2) - 8 =? 0 (check)

Pick any value for α, determine the values for x, y, z depending on α, and you will see that both equations are true.

Another way to specify the solution set is as an ordered triple:

(x, 1/2{21x-25), 4x) depending on x.

Answer 2

when given two equations three variables, can often find solution y=f(x), y=f(z), or x=f(z).

here z=((1/21)(8y+100). Let y be the parameter µ, then z=(1/21)(8µ+100), and x=(1/84)(8µ+100)