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qwelly [4]
3 years ago
8

) The points (1, 0, 3), (1, 1, 1), and (−2, −1, 2) lie on a unique plane a1x1 + a2x2 + a3x3 = b. Using your previous answers, fi

nd an equation for this plane. (Hint: think about the relationship between the previous system and the one you would need to solve in this question.)

Mathematics
2 answers:
Rudik [331]3 years ago
6 0

Answer:

x1-2x2-x3= -2

Step-by-step explanation:

Substitute the three points to get three equations:

a1+0a2+ 3a3=d --------------eq i

a1+a2+a3=d----------------eq ii

-2a1-a2+2a3=d---------------eq iii

subtract eq i from eq ii

a2-2a3=0 ----------- eq iv

now add eq iv to eq iii

-2a1=d ------- eq v

put eq v in eq i

a1 + 3a3=-2a1

a3=  -a1 ----------- eq vi

put eq vi in eq iv

a2-2(-a1)=0

a2= -2a1 ------ eq vii

put eq v, eq vi and eq vii in the equation of plane

a1x1+ a2x2+a3x3=d

a1x1 -2a1x2-a1x3=-2a1

x1-2x2-x3= -2

Akimi4 [234]3 years ago
6 0

Answer:

2x₁ - x₂ + x₃ = 2

Question in full

(a) Find the general solution for the following system of linear equations:

z2 + 3z3 − z4 = 0

−z1 − z2 − z3 + z4 = 0

−2z1 − 4z2 + 4z3 − 2z4 = 0

(b) Give an example of a solution to the previous system of linear equations.

(c) The points (0, 1, 3), (1, 1, 1), and (−1, −2, 2) lie on a unique plane in R 3 , defined by an equation of the form a1x1 + a2x2 + a3x3 = b. Using your previous answers, find an equation for this plane. (Hint: think about the relationship between the previous system and the one you would need to solve in this question.)

Step-by-step explanation:

The augmented matrix is \left[\begin{array}{ccc}1&2&3&0\\4&5&6&0\\7&8&9&0\end{array}\right]

The echelon form

The general solution and the sample solution

as well as the equation of the plane

are given in the attached solution below

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The table shows some values of a function of the form y = ax2 + bx + c.
natali 33 [55]

Answer:

Value of constant term c is (-4)

Step-by-step explanation:

The given table represents a function which is in the form of a quadratic equation,

y = ax² + bx + c

We choose three points (3, -10), (4, -16) and (5, -24) from the table and satisfy the equation to get the values of a, b, and c.

For point (3, -10)

-10 = a(3)² + 3b + c

9a + 3b + c = -10 -------(1)

For point (4, -16)

-16 = a(4)² + 4b + c

16a + 4b + c = -16 ------(2)

For point (5, -24)

-24 = a(5)² + 5b + c

25a + 5b + c = -24 -----(3)

Equation (1) - equation (2)

(9a + 3b + c) - (16a + 4b + c) = -10 + 16

-7a - b = 6

7a + b = -6 ------(4)

Equation (2) - equation (3)

(16a + 4b + c) - (25a + 5b + c) = -16 + 24

-9a - b = 8

9a + b = -8 -------(5)

Equation (4) - Equation (5)

(7a + b) - (9a + b) = -6 + 8

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From equation (4),

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b = -6 + 7

b = 1

From equation (1)

9(-1) + 3(1) + c = -10

-9 + 3 + c = -10

c = -10 + 6

c = -4

Therefore, the value of constant term c is (-4).

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