Show that the following quations are consistent : 5x + 3y + 14z = 4; y + 2z = l ;x - y + 2z = 0 ; 2x + y + 6z = 2. .
Answers
The set of values x = l/5 = 0/5 = 0, y = -1/5 + 0 = -1/5, and z = 1/5 satisfies all of the given equations simultaneously, and the equations are consistent.
Given:-
5x + 3y + 14z = 4
y + 2z = l
x - y + 2z = 0
2x + y + 6z = 2
To Find:-
Prove that the following equations are consistent
Solution:-
To show that the given equations are consistent, we need to find a set of values for x, y, and z that satisfy all of the equations simultaneously. To do this, we can use the method of substitution.
First, we can solve the second equation for y: y = -2z + l. Substituting this expression into the first and third equations, we get:
⇒5x - 6z + l + 14z = 4
⇒x + z + l = 0
Next, we can solve the first equation for z: z = (4 - 5x - l)/14. Substituting this expression into the second equation, we get:
⇒-2((4 - 5x - l)/14) + l = -2(4 - 5x - l)/14 + l = 0
Solving for x, we find that x = l/5. Substituting this value into the first equation, we get:
⇒5(l/5) - 6z + l + 14z = 4
⇒l - 6z + l + 14z = 4
⇒15z = 3
⇒z = 1/5
Finally, substituting this value of z into the expression for y, we find that y = -2(1/5) + l = -1/5 + l. Substituting these values of x, y, and z into the third equation, we get:
⇒l/5 + 1/5 + l = 0
⇒(6/5)l = 0
⇒l = 0
∴ the set of values x = l/5 = 0/5 = 0, y = -1/5 + 0 = -1/5, and z = 1/5 satisfies all of the given equations simultaneously, and the equations are consistent.
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