The value of a is 1/4 and the value of h and k is (2,-7/4)
<h3>What is a Parabola ?</h3>
A parabola is a u shaped curve. It is a plane curve whose all points are at a fixed distance form a point called focus.
(x-h)² = 4a(y-k)
It is given that the parabola goes through the point (3, -1), and the vertex is at (2, -2).
Therefore
(x -2)² = 4a(y +2)
The parabola passes through the point (3,-1)
(3-2)² = 4*a(-1+2)
1 = 4 a
a = 1/4
Now to determine the value of focus point , (h,k)
(h = 2)
k = - 2 +1/4 = -7/4
Therefore The value of a is 1/4 and the value of h and k is (2,-7/4)
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Answer:
x=-1/85; y=-283/85; z=2/17
Step-by-step explanation:
Using an algebraic method like elimination or substitution would take a lot of steps which could lead to mistake the calculations. In this case, I decided to use the Gaussian elimination. We can express the system in matrix form as follows:
![\left[\begin{array}{ccc}2&-4&6\\9&-3&1\\5&0&9\end{array}\right] \left[\begin{array}{ccc}x\\y\\z\end{array}\right] =\left[\begin{array}{ccc}14\\10\\1\end{array}\right]](https://tex.z-dn.net/?f=%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D2%26-4%266%5C%5C9%26-3%261%5C%5C5%260%269%5Cend%7Barray%7D%5Cright%5D%20%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7Dx%5C%5Cy%5C%5Cz%5Cend%7Barray%7D%5Cright%5D%20%3D%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D14%5C%5C10%5C%5C1%5Cend%7Barray%7D%5Cright%5D)
To begin the calculations, we write the system in augmented matrix form and use the Gaussian elimination:
![\left[\begin{array}{ccccc}2&-4&6&|&14\\9&-3&1&|&10\\5&0&9&|&1\end{array}\right]](https://tex.z-dn.net/?f=%5Cleft%5B%5Cbegin%7Barray%7D%7Bccccc%7D2%26-4%266%26%7C%2614%5C%5C9%26-3%261%26%7C%2610%5C%5C5%260%269%26%7C%261%5Cend%7Barray%7D%5Cright%5D)
By applying the Gaussian elimination, the final matrix is the following:
![\left[\begin{array}{ccccc}1&0&0&|&-1/85\\0&1&0&|&-283/85\\0&0&1&|&2/17\end{array}\right]](https://tex.z-dn.net/?f=%5Cleft%5B%5Cbegin%7Barray%7D%7Bccccc%7D1%260%260%26%7C%26-1%2F85%5C%5C0%261%260%26%7C%26-283%2F85%5C%5C0%260%261%26%7C%262%2F17%5Cend%7Barray%7D%5Cright%5D)
In order to verify the results, it´s enough to substitute the calculated values in the original equations to see if the equalities are correct. Here you can see the verification for all of the equations:
Answer:
65 < or equal to x > or equal to 100
Ps: had to put "or equal to", because I can't put the lines underneath
Anyways, when an inequality says and, it is automatically a compound inequality. The x is in the middle.
2x - 2y + z = 5....multiply by -1
2x - 2y + 3z = 7
-------------------
-2x + 2y - z = -5 (result of multiplying by -1)
2x - 2y + 3z = 7
-------------------add
2z = 2
z = 2/2 = 1
2x - 2y + z = 5.....multiply by 3
-3x - 3y + 3z = 9...multiply by 2
-------------------
6x - 6y + 3z = 15 (result of multiplying by 3)
-6x - 6y + 6z = 18 (result of multiplying by 2)
-------------------add
-12y + 9z = 33
-12y + 9(1) = 33
-12y + 9 = 33
-12y = 33 - 9
-12y = 24
y = -24/12
y = -2
2x - 2y + z = 5
2x - 2(-2) + 1 = 5
2x + 4 + 1 = 5
2x + 5 = 5
2x = 5 - 5
2x = 0
x = 0
solution is : (0,-2,1)
Answer:
(-3, -2) is a solution of y = -2x + -8
Step-by-step explanation:
y = -2x + -8
(-3, -2) are the coordinates of P(x,y)
-2= -2(-3) - 8
-2 = +6 - 8
-2 = -2
Since LHS = RHS, (-3, -2) is a solution of y = -2x + -8
