SOLUTION: Solve the following Using Cramer's rule 1.) 8x+5y=7 6x-7y=59 2.) 3x-7y=34 8x+9y=-20 3.) 3x-2y+2z=0 4x+3y-z=-15 2x+5y+4z=-10

Algebra ->  Quadratic Equations and Parabolas  -> Quadratic Equations Lessons  -> Quadratic Equation Lesson -> SOLUTION: Solve the following Using Cramer's rule 1.) 8x+5y=7 6x-7y=59 2.) 3x-7y=34 8x+9y=-20 3.) 3x-2y+2z=0 4x+3y-z=-15 2x+5y+4z=-10      Log On


   

Question 474514: Solve the following Using Cramer's rule
1.) 8x+5y=7
6x-7y=59
2.) 3x-7y=34
8x+9y=-20
3.) 3x-2y+2z=0
4x+3y-z=-15
2x+5y+4z=-10
Answer by MathLover1(20849) About Me  (Show Source):
You can put this solution on YOUR website!

1.) 8x%2B5y=7
+6x-7y=59

Solved by pluggable solver: Using Cramer's Rule to Solve Systems with 2 variables



system%288%2Ax%2B5%2Ay=7%2C6%2Ax%2B-7%2Ay=59%29



First let A=%28matrix%282%2C2%2C8%2C5%2C6%2C-7%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 59 which are highlighted here:
system%288%2Ax%2B5%2Ay=highlight%287%29%2C6%2Ax%2B-7%2Ay=highlight%2859%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=%288%29%28-7%29-%285%29%286%29=-86. 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%2C5%2Chighlight%2859%29%2C-7%29%29



Now compute the determinant of A%5Bx%5D to get abs%28A%5Bx%5D%29=%287%29%28-7%29-%285%29%2859%29=-344. 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=%28-344%29%2F%28-86%29=4



So the first solution is x=4




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


We'll follow the same basic idea to find the other solution. Let's reset by letting A=%28matrix%282%2C2%2C8%2C5%2C6%2C-7%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%2C8%2Chighlight%287%29%2C6%2Chighlight%2859%29%29%29



Now compute the determinant of A%5By%5D to get abs%28A%5By%5D%29=%288%29%2859%29-%287%29%286%29=430.



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=%28430%29%2F%28-86%29=-5



So the second solution is y=-5




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

Final Answer:




So the solutions are x=4 and y=-5 giving the ordered pair (4, -5)




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






2.) 3x-7y=34
8x%2B9y=-20
Solved by pluggable solver: Using Cramer's Rule to Solve Systems with 2 variables



system%283%2Ax%2B-7%2Ay=34%2C8%2Ax%2B9%2Ay=-20%29



First let A=%28matrix%282%2C2%2C3%2C-7%2C8%2C9%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 34 and -20 which are highlighted here:
system%283%2Ax%2B-7%2Ay=highlight%2834%29%2C8%2Ax%2B9%2Ay=highlight%28-20%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=%283%29%289%29-%28-7%29%288%29=83. 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%2834%29%2C-7%2Chighlight%28-20%29%2C9%29%29



Now compute the determinant of A%5Bx%5D to get abs%28A%5Bx%5D%29=%2834%29%289%29-%28-7%29%28-20%29=166. 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=%28166%29%2F%2883%29=2



So the first solution is x=2




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


We'll follow the same basic idea to find the other solution. Let's reset by letting A=%28matrix%282%2C2%2C3%2C-7%2C8%2C9%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%2C3%2Chighlight%2834%29%2C8%2Chighlight%28-20%29%29%29



Now compute the determinant of A%5By%5D to get abs%28A%5By%5D%29=%283%29%28-20%29-%2834%29%288%29=-332.



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=%28-332%29%2F%2883%29=-4



So the second solution is y=-4




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

Final Answer:




So the solutions are x=2 and y=-4 giving the ordered pair (2, -4)




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




3.) 3x-2y%2B2z=0
4x%2B3y-z=-15
2x%2B5y%2B4z=-10
Solved by pluggable solver: Using Cramer's Rule to Solve Systems with 3 variables







First let A=%28matrix%283%2C3%2C3%2C-2%2C2%2C4%2C3%2C-1%2C2%2C5%2C4%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 0, -15, and -10 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=115. 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=-230. 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=%28-230%29%2F%28115%29=-2



So the first solution is x=-2




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


We'll follow the same basic idea to find the other two solutions. Let's reset by letting A=%28matrix%283%2C3%2C3%2C-2%2C2%2C4%2C3%2C-1%2C2%2C5%2C4%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=-230.



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=%28-230%29%2F%28115%29=-2



So the second solution is y=-2




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





Let's reset again by letting A=%28matrix%283%2C3%2C3%2C-2%2C2%2C4%2C3%2C-1%2C2%2C5%2C4%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=115.



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=%28115%29%2F%28115%29=1



So the third solution is z=1




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

Final Answer:




So the three solutions are x=-2, y=-2, and z=1 giving the ordered triple (-2, -2, 1)




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.