x + 2y - z = -5 (Eqn 1)

2x - 3y + 4z = 28 (Eqn 2)

4x + 5y - 3z = -10 (Eqn 3)

I will not eliminate x as the hint suggested but leave that to you. I will remove the z instead.

Multiply Eqn 1 by 4:

4x + 8y - 4z = -20 (Eqn 4)

Add Eqn 2 to the Eqn 4:

6x + 5y = 8 (Eqn 5)

Multiply Eqn 1 by 3:

3x + 6y - 3z = -15 (Eqn 6)

Subtract Eqn 6 from Eqn 3:

x - y = 5 (Eqn 7)

Multiply this eqn by 5 and add
to the other new equation:

11x = 33

x = 3

So y = 3 - 5 = -2

3 - 4 - z = -5

z = 4.

So x = 3, y = -2 and z = 4.

YOU substitute these values into one of the three equations to verify these solutions.

e - f + g = 10 (Eqn 1)

4e + 2f - 3g = 8 (Eqn 2)

3e - 5f + 2g = 34 (Eqn 3)

Eqn 1 × 3
3e - 3f + 3g = 30 (Eqn 4)

Add Eqn 2 and Eqn 4:

7e - f = 38 (Eqn 5)

Eqn 1 × 2

2e - 2f + 2g = 20 (Eqn 6)

Subtract Eqn 6 from Eqn 3

e - 3f = 14 (Eqn 7)

Now substitute for e in Eqn 5
using Eqn 7:

7(3f + 14) - f = 38

20f = -60

f = -3

So e = 14 + 3(-3) = 5

Substitute for e and f in Eqn 1:

5 -(-3) +g = 10

g = 2

So e = 5, f = -3 and g = 2.

6x + 4y - 2z = 0 (Eqn 1)

3x - 2y + 4z = 3 (Eqn 2)

5x - 2y + 6z = 3 (Eqn 3)

(Eqn 3 - Eqn 2)

2x + 2z = 0 (Eqn 4)

(Eqn 2 ×2)

6x - 4y + 8z = 6 (Eqn 5)

Eqn 1 plus Eqn 5

12x + 6z = 6 (Eqn 6)

Divide by 6:

2x + z = 1 (Eqn 7)

Eqn 4 - Eqn 7

z = -1

So x = 1

Substitute into Eqn 1:

6(1) + 4y -2(-1) = 0

4y = -8

y = -2

So x = 1, y = -2 and z = -1.

2a - b - c = 9 (Eqn 1)

5a + 2c = -3 (Eqn 2)

7a - 2b = 1 (Eqn 3)

Eqn 3 only has a and b terms so we could eliminate c from the other 2 eqns:

Eqn 1 × 2
4a - 2b - 2c = 18 (Eqn 4)

Add Eqn 2 and Eqn 4:

9a - 2b = 15 (Eqn 5)

Subtract Eqn 5 from Eqn 3

2a = 14

a = 7

Now substitute for a in Eqn 3:

b = 24

Now substitute for a and b in Eqn 1:

2(7) - 24 - c = 9

c = -19

So a = 7, b = 24 and c = -19.

3p - q - 2r = -15 (Eqn 1)

20p - 3q - 5r = -15 (Eqn 2)

5p + 2q + 3r = -16 (Eqn 3)

q is the variable which has the lowest coefficients and so it is esiest to eliminate.

Eqn 1 × 2
6p - 2q - 4r = -30 (Eqn 4)

Add Eqn 3 and Eqn 4:

11p - r = -46 (Eqn 5)

Eqn 1 × 3

9p - 3q - 6r = -45 (Eqn 6)

Subtract Eqn 6 from Eqn 2

11p + r = 30 (Eqn 7)

Add Eqn 5 to Eqn 7:

22p = -16

Using Eqn 7:

Using Eqn 1:

x - 2y = 3 (Eqn 1)

4y - 3z = 4 (Eqn 2)

x + 3z = 2 (Eqn 3)

Subtract Eqn 1 from Eqn 3
2y + 3z = -1 (Eqn 4)

Add Eqn 2 and Eqn 4:

6y = 3

y = 0.5

Substitute into Eqn 1

x - 2(0.5) = 3

x = 4

Substitute into Eqn 3

4 + 3z = 2

1. Substituting into y = k(x - a)² + b

(-3, 7):

7 = k(-3-a)² + b

7 = 9k + 6ak + a²k + b (eqn 1)

(0, -5):

-5 = k(-a)² + b

-5 = ka² + b (eqn 2)

(2, -3):

-3 = k(2 - a)²2 + b

-3 = 4k - 4ak + a²k + b (eqn 3)

Eqn 1 - eqn 2:

12 = 9k + 6ak

4 = 3k + 2ak (eqn 4)

Eqn 1 - eqn 3:

10 = 5k + 10ak

2 = k + 2ak (eqn 5)

Eqn 4 - eqn 5:

2 = 2k

k = 1

Substituting into eqn 5:

2 = 1 + 2a

a = ½

Substituting into eqn 2:

-5 = 1 × ¼ + b

b = .

∴ Equation of the parabola is