SOLUTION: A chemist mixes two different solutions with concentrations of 20% and 40%HCL to obtain 21 L of 33 1/3 HCL. What will represent the amount of 20% solution used according to Cramer'

Algebra ->  Matrices-and-determiminant -> SOLUTION: A chemist mixes two different solutions with concentrations of 20% and 40%HCL to obtain 21 L of 33 1/3 HCL. What will represent the amount of 20% solution used according to Cramer'      Log On


   

Question 888424: A chemist mixes two different solutions with concentrations of 20% and 40%HCL to obtain 21 L of 33 1/3 HCL. What will represent the amount of 20% solution used according to Cramer's Rule?
Answer by richwmiller(17219) About Me  (Show Source):
You can put this solution on YOUR website!
we want x
Solved by pluggable solver: Using Cramer's Rule to Solve Systems with 2 variables



system%280.2%2Ax%2B0.4%2Ay=7%2C1%2Ax%2B1%2Ay=21%29



First let A=%28matrix%282%2C2%2C0.2%2C0.4%2C1%2C1%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 7 and 21 which are highlighted here:
system%280.2%2Ax%2B0.4%2Ay=highlight%287%29%2C1%2Ax%2B1%2Ay=highlight%2821%29%29



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=%280.2%29%281%29-%280.4%29%281%29=-0.2. Remember that the determinant of the 2x2 matrix A=%28matrix%282%2C2%2Ca%2Cb%2Cc%2Cd%29%29 is abs%28A%29=ad-bc. If you need help with calculating the determinant of any two by two matrices, then 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).


A%5Bx%5D=%28matrix%282%2C2%2Chighlight%287%29%2C0.4%2Chighlight%2821%29%2C1%29%29



Now compute the determinant of A%5Bx%5D to get abs%28A%5Bx%5D%29=%287%29%281%29-%280.4%29%2821%29=-1.4. Once again, remember that the determinant of the 2x2 matrix A=%28matrix%282%2C2%2Ca%2Cb%2Cc%2Cd%29%29 is abs%28A%29=ad-bc



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=%28-1.4%29%2F%28-0.2%29=7



So the first solution is x=7




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


We'll follow the same basic idea to find the other solution. Let's reset by letting A=%28matrix%282%2C2%2C0.2%2C0.4%2C1%2C1%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).


A%5Bx%5D=%28matrix%282%2C2%2C0.2%2Chighlight%287%29%2C1%2Chighlight%2821%29%29%29



Now compute the determinant of A%5By%5D to get abs%28A%5By%5D%29=%280.2%29%2821%29-%287%29%281%29=-2.8.



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=%28-2.8%29%2F%28-0.2%29=14



So the second solution is y=14




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

Final Answer:




So the solutions are x=7 and y=14 giving the ordered pair (7, 14)




Once again, Cramer's Rule is dependent on determinants. Take a look at this 2x2 Determinant Solver if you need more practice with determinants.