If a(0,-1,3)+b(1,1,1)-c(1,2,5)=(-2,3,-8), determine a,b,c
Is this the correct answer: a= -4, b= -3, c= -1?
0a+1b-1c = -2
-a+1b-2c = 3
3a+1b-5c = -10
has solutions (-4,-3,-1).
Is there a typo in your question?
As written, the solutions are (-26/7, -23/7, -9/7)
To determine the values of a, b, and c in the equation a(0,-1,3) + b(1,1,1) - c(1,2,5) = (-2,3,-8), we can equate the corresponding components of the vectors on both sides of the equation.
Let's first equate the x-components:
0 + b - c = -2 (equating the x-components)
Now, equating the y-components:
-1 + b - 2c = 3 (equating the y-components)
Lastly, equating the z-components:
3 + b - 5c = -8 (equating the z-components)
Now, we have a system of linear equations that we can solve to find the values of a, b, and c.
To solve this system, we can use different methods such as substitution or elimination. Let's use the method of elimination:
By adding the first and second equation, we can eliminate 'b':
0 + b - c + (-1 + b - 2c) = -2 + 3
Simplifying:
2b - 3c - 1 = 1
Rearranging:
2b - 3c = 2
Next, let's add the second and third equation to eliminate 'b':
-1 + b - 2c + (3 + b - 5c) = 3 - 8
Simplifying:
2b - 7c + 2 = -5
Rearranging:
2b - 7c = -7
Now, we have a system of two equations with two variables:
2b - 3c = 2
2b - 7c = -7
To solve this system, we can subtract the first equation from the second equation:
(2b - 7c) - (2b - 3c) = -7 - 2
Simplifying:
(2b - 7c) - (2b - 3c) = -9
Simplifying further:
-4c = -9
Dividing both sides by -4 gives:
c = 9/4
Now, we can substitute the value of c back into one of the earlier equations to find the value of b:
2b - 7(9/4) = -7
8b - 63 = -28
8b = 63 - 28
8b = 35
b = 35/8
Finally, we substitute the values of b and c back into one of the earlier equations to find the value of a:
2(35/8) - 3(9/4) = 2
70/8 - 27/4 = 2
35/4 - 27/4 = 2
8/4 = 2
a = 2
Therefore, the correct values of a, b, and c are:
a = 2
b = 35/8
c = 9/4