Find values of a, b, and c which satisfies the following equation:
a(2,1,0) + b(-3,4,5) + c(2,0,3)= (-4,10,7)
You will have to solve 3 equations in 3 unknowns
2a - 3b + 2c = -4
a + 4b = 10
5b + 3c = 7
I got that far i just wasn't sure how to solve for 3 different unknowns unless I represent some unknown as the same as another unknown in the question?
double the 2nd and subtract from the first
-11b + 2c = -24 (#4)
#4x 3 : ---> -33b + 6c = -72
#3x 2 : ---> 10b + 6c = 14
subtract them
-43b = -86
b = 2
in #4 : -22 + 2c = -24 ,
c = -1
in #2 : a + 8 = 10
a = 2
a=2, b=2, c=-1
check:
2(2,1,0) + 2(-3,4,5) - 1(2,0,3)
= (-4,10,7)
= RS
great thanks :)
To find the values of a, b, and c that satisfy the given equation, we need to equate the corresponding components of both sides of the equation.
Given equation:
a(2,1,0) + b(-3,4,5) + c(2,0,3) = (-4,10,7)
Equating the components, we get three equations:
For the x-components:
2a - 3b + 2c = -4 ............(1)
For the y-components:
a + 4b = 10 ............(2)
For the z-components:
5b + 3c = 7 ............(3)
Now we have a system of three linear equations with three variables (a, b, c). We can solve this system of equations to find the values of a, b, and c.
One way to do this is by using the method of substitution or elimination. Let's use the method of substitution to solve this system of equations.
From equation (2), we can solve for 'a' in terms of 'b':
a = 10 - 4b
Substituting this value of 'a' in equations (1) and (3), we get:
Equation (1):
2(10 - 4b) - 3b + 2c = -4
20 - 8b - 3b + 2c = -4
-11b + 2c = -24 ............(4)
Equation (3):
5b + 3c = 7 ............(5)
Now, we have two equations, (4) and (5), with two variables, 'b' and 'c'. We can solve this system of equations to find the values of 'b' and 'c'.
Multiplying equation (4) by 5 and equation (5) by 11 to eliminate 'b', we get:
-55b + 10c = -120 ............(6)
55b + 33c = 77 ............(7)
Adding equations (6) and (7), we can eliminate 'b', and solve for 'c':
-55b + 10c + 55b + 33c = -120 + 77
43c = -43
c = -1
Now we substitute the value of 'c' back into equation (5) to solve for 'b':
5b + 3(-1) = 7
5b - 3 = 7
5b = 7 + 3
5b = 10
b = 2
Finally, we substitute the values of 'b' and 'c' into equation (2) to find 'a':
a + 4(2) = 10
a + 8 = 10
a = 10 - 8
a = 2
Therefore, the values of a, b, and c that satisfy the given equation are:
a = 2
b = 2
c = -1