Find the values of the variables.
[-12 -w^2] = [2k -81]
[2f 3] [-4 3]
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Looks like
k = -6
w = 9 or -9
f = -2
Well, first let's break down the equation into two separate equations:
-12 = 2k - 81 (Equation 1)
-w^2 = -4 (Equation 2)
2f = -4 (Equation 3)
3 = 3 (Equation 4)
Now, let's solve each equation one by one:
From Equation 1:
-12 + 81 = 2k
69 = 2k
k = 69/2
k = 34.5
From Equation 2:
-w^2 = -4
w^2 = 4
w = 2 or w = -2
From Equation 3:
2f = -4
f = -4/2
f = -2
From Equation 4:
We already know that 3 = 3, so no additional information can be derived from this equation.
So, the values of the variables are:
k = 34.5,
w = 2 or -2,
f = -2
To find the values of the variables, we can compare the corresponding elements in each matrix.
From the first row, we have:
-12 = 2k - 81
Simplifying the equation, we get:
2k = -12 + 81
2k = 69
Dividing both sides by 2, we find:
k = 69/2
k = 34.5
From the second row, we have:
-2f = -4
Dividing both sides by -2, we get:
f = 2
Combining both results, we have:
k = 34.5
f = 2
To find the values of the variables, we can set up a system of equations by equating the corresponding elements in the matrices. Let's go step by step:
First, equate the elements in the first column:
-12 = 2k - 81 (Equation 1)
2f = -4 (Equation 2)
Next, equate the elements in the second column:
-w^2 = 3 (Equation 3)
3 = 3 (Equation 4)
From Equation 4, we can see that 3 = 3, which is always true. This means the second equation, Equation 2, is satisfied regardless of the value of f. So, we can ignore Equation 2 for now.
Now, let's solve Equation 3 for w:
-w^2 = 3
Multiplying both sides by -1, we get:
w^2 = -3
Since w^2 is equal to a negative number, there is no real solution for w. This means we cannot determine the value of w.
Moving on to Equation 1:
-12 = 2k - 81
Add 81 to both sides:
69 = 2k
Divide both sides by 2:
k = 34.5
So, the value of k is 34.5.
In summary:
w has no real solution.
k = 34.5
Please note that we were not able to determine the value of f since we ignored Equation 2.