Using the Gauss-Jordan method need to determine the coefficients a,b,c of the quadratic equation
x=ax squared +bx+c y=5.4 and x=8
Don't you mean
y = ax^2 +bx + c ?
You would need three (x,y) pairs to determine three coefficients. You only provided one.
The Gauss-Jordan method applies to matrices of more than one linear equation. You have one quadratic equation.
I (or the question) must be missing something, but I don't see it.
Here are the three equations
x=8, x=13, x=18 y=5.4, y=6.3, y=5.6
Yes I do mean y = ax^2 +bx + c
<<Here are the three equations
x=8, x=13, x=18 y=5.4, y=6.3, y=5.6
>>
Those are not three equations. They are possibly the coordinates of three points, arranged in a confusing manner.
(x,y) = (8,5.4), (13,6.3) and (5.5,5.6)
With those points, and the Gauss-Jordan method, it is possible to calculate the values of a, b, and c for a parabolic (quadratic) curve
y = ax^2 + bx +c gthat goes through through those points. You would be solving these 3 equations:
64a + 8b + c = 5.4
169a + 13b + c = 6.3
30.25a + 5.5b + c = 5.6
The next step is to evalauate four determinants. One of them is
|64 8 1|
|169 13 1|
|30.25 5.5 1|
That is as far as I am going to go with this.
I can solve from here. My problem is I cannot pick out the data and put it in the right order. Any suggestions.
To determine the coefficients of the quadratic equation using the Gauss-Jordan method, we need to set up a system of equations using the given points.
First, let's substitute the given values into the equation:
When y = 5.4 and x = 8:
5.4 = a(8^2) + b(8) + c
Now, we can rewrite this equation in standard form:
a(64) + b(8) + c = 5.4
Now, let's set up a system of equations based on different powers of x:
1. Equation 1: a(64) + b(8) + c = 5.4
2. Equation 2: a(1^2) + b(1) + c = y1 (where x = 1 and y = y1)
3. Equation 3: a(0^2) + b(0) + c = y0 (where x = 0 and y = y0)
We can rewrite these equations in matrix form:
| 64 8 1 | | a | | 5.4 |
| 1 1 1 | x | b | = | y1 |
| 0 0 1 | | c | | y0 |
Now, we need to perform row operations using the Gauss-Jordan method to solve for the coefficient matrix [a, b, c].
Step 1: Swap rows if necessary to bring 64 to the top left corner of the matrix.
| 64 8 1 | | a | | 5.4 |
| 1 1 1 | x | b | = | y1 |
| 0 0 1 | | c | | y0 |
Step 2: Divide row 1 by 64 to create leading 1.
| 1 1/8 1/64 | | a | | 5.4 |
| 1 1 1 | x | b | = | y1 |
| 0 0 1 | | c | | y0 |
Step 3: Replace row 2 by the difference between row 2 and row 1.
| 1 1/8 1/64 | | a | | 5.4 |
| 0 7/8 63/64 | x | b | = | y1 - 5.4 |
| 0 0 1 | | c | | y0 |
Step 4: Replace row 2 by the difference between 8 * row 2 and row 3.
| 1 1/8 1/64 | | a | | 5.4 |
| 0 7/8 0 | x | b | = | (y1 - 5.4) - (63/64)y0 |
| 0 0 1 | | c | | y0 |
Step 5: Divide row 2 by (7/8) to create leading 1.
| 1 1/8 1/64 | | a | | 5.4 |
| 0 1 0 | x | b | = | (y1 - 5.4) - (63/64)y0 * (8/7) |
| 0 0 1 | | c | | y0 |
Step 6: Replace row 1 by the difference between row 1 and (1/8) * row 2.
| 1 0 1/64 | | a | | 5.4 - (1/8)((y1 - 5.4) - (63/64)y0 * (8/7)) |
| 0 1 0 | x | b | = | (y1 - 5.4) - (63/64)y0 * (8/7) |
| 0 0 1 | | c | | y0 |
Step 7: Replace row 1 by the difference between row 1 and (1/64) * row 3.
| 1 0 0 | | a | | 5.4 - (1/64)y0 |
| 0 1 0 | x | b | = | (y1 - 5.4) - (63/64)y0 * (8/7) |
| 0 0 1 | | c | | y0 |
Therefore, the coefficients of the quadratic equation are:
a = 5.4 - (1/64)y0
b = (y1 - 5.4) - (63/64)y0 * (8/7)
c = y0
To get the specific values of a, b, and c, you need to substitute the given values of y1 and y0 into the equations.