SOLUTION: what is the value of a,b,c 2a-5b+c=1 a+c=2 b-3c=-3

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Question 899057: what is the value of a,b,c
2a-5b+c=1
a+c=2
b-3c=-3
Answer by ewatrrr(24785) About Me  (Show Source):
You can put this solution on YOUR website!
 

Hi

2a-5b+c=1, c = 2-a  and b = -3 + 3c

2a - 5(-3+3(2-a)) + (2-a)= 1

2a - 5(3 - 3a) + 2 - a = 1

2a - 15 + 15a + 2 - a = 1

  16a = 14

    a = 14/16 = 7/8 

    c = 16%2F8+-+7%2F8 = 9/8

    b = -3+%2B+27%2F8+=+-24%2F8+%2B+27%2F8 = 3/8

OR




 
Solved by pluggable solver: Using Cramer's Rule to Solve Systems with 3 variables




  

  

  system%282%2Ax%2B-5%2Ay%2B1%2Az=1%2C1%2Ax%2B0%2Ay%2B1%2Az=2%2C0%2Ax%2B1%2Ay%2B-3%2Az=-3%29

  

  

  

  First let A=%28matrix%283%2C3%2C2%2C-5%2C1%2C1%2C0%2C1%2C0%2C1%2C-3%29%29. This is the matrix formed by the coefficients of the given system of equations.

  

  

  Take note that the right hand values of the system are 1, 2, and -3 and they are highlighted here: 

  

  

  

  

  These values are important as they will be used to replace the columns of the matrix A.

  

  

  

  

  Now let's calculate the the determinant of the matrix A to get abs%28A%29=-16. To save space, I'm not showing the calculations for the determinant. However, if you need help with calculating the determinant of the matrix A, check out this solver.

  

  

  

  Notation note: abs%28A%29 denotes the determinant of the matrix A.

  

  

  

  ---------------------------------------------------------

  

  

  

  Now replace the first column of A (that corresponds to the variable 'x') with the values that form the right hand side of the system of equations. We will denote this new matrix A%5Bx%5D (since we're replacing the 'x' column so to speak).

  

  

  

  

  

  

  Now compute the determinant of A%5Bx%5D to get abs%28A%5Bx%5D%29=-14. Again, as a space saver, I didn't include the calculations of the determinant. Check out this solver to see how to find this determinant.

  

  

  

  To find the first solution, simply divide the determinant of A%5Bx%5D by the determinant of A to get: x=%28abs%28A%5Bx%5D%29%29%2F%28abs%28A%29%29=%28-14%29%2F%28-16%29=7%2F8

  

  

  

  So the first solution is x=7%2F8

  

  

  

  

  ---------------------------------------------------------

  

  

  We'll follow the same basic idea to find the other two solutions. Let's reset by letting A=%28matrix%283%2C3%2C2%2C-5%2C1%2C1%2C0%2C1%2C0%2C1%2C-3%29%29 again (this is the coefficient matrix).

  

  

  

  

  Now replace the second column of A (that corresponds to the variable 'y') with the values that form the right hand side of the system of equations. We will denote this new matrix A%5By%5D (since we're replacing the 'y' column in a way).

  

  

  

  

  

  

  Now compute the determinant of A%5By%5D to get abs%28A%5By%5D%29=-6.

  

  

  

  To find the second solution, divide the determinant of A%5By%5D by the determinant of A to get: y=%28abs%28A%5By%5D%29%29%2F%28abs%28A%29%29=%28-6%29%2F%28-16%29=3%2F8

  

  

  

  So the second solution is y=3%2F8

  

  

  

  

  ---------------------------------------------------------

  

  

  

  

  

  Let's reset again by letting A=%28matrix%283%2C3%2C2%2C-5%2C1%2C1%2C0%2C1%2C0%2C1%2C-3%29%29 which is the coefficient matrix.

  

  

  

  Replace the third column of A (that corresponds to the variable 'z') with the values that form the right hand side of the system of equations. We will denote this new matrix A%5Bz%5D 

  

  

  

  

  

  

  Now compute the determinant of A%5Bz%5D to get abs%28A%5Bz%5D%29=-18.

  

  

  

  To find the third solution, divide the determinant of A%5Bz%5D by the determinant of A to get: z=%28abs%28A%5Bz%5D%29%29%2F%28abs%28A%29%29=%28-18%29%2F%28-16%29=9%2F8

  

  

  

  So the third solution is z=9%2F8

  

  

  

  

  ====================================================================================

  

  Final Answer:

  

  

  

  

  So the three solutions are x=7%2F8, y=3%2F8, and z=9%2F8 giving the ordered triple (7/8, 3/8, 9/8)

  

  

  

  

  Note: there is a lot of work that is hidden in finding the determinants. Take a look at this 3x3 Determinant Solver to see how to get each determinant.