SOLUTION: solve the following system of equation. enter the y- coordinate of the solution 5x+2y=21 -2x+6y=-34

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Question 863813: solve the following system of equation. enter the y- coordinate of the solution
5x+2y=21
-2x+6y=-34
Answer by ewatrrr(24785) About Me  (Show Source):
You can put this solution on YOUR website!


Hi

 5x+2y=21   |multiplying thru by -3 to  eliminate the y variable by adding the EQs

-2x+6y=-34

  -17x = -97

     x = 97/17,  y = -64/17




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




  

  

  system%285%2Ax%2B2%2Ay=21%2C-2%2Ax%2B6%2Ay=-34%29

  

  

  

  First let A=%28matrix%282%2C2%2C5%2C2%2C-2%2C6%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 21 and -34 which are highlighted here: 

  system%285%2Ax%2B2%2Ay=highlight%2821%29%2C-2%2Ax%2B6%2Ay=highlight%28-34%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=%285%29%286%29-%282%29%28-2%29=34. 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%2821%29%2C2%2Chighlight%28-34%29%2C6%29%29

  

  

  

  Now compute the determinant of A%5Bx%5D to get abs%28A%5Bx%5D%29=%2821%29%286%29-%282%29%28-34%29=194. 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=%28194%29%2F%2834%29=97%2F17

  

  

  

  So the first solution is x=97%2F17

  

  

  

  

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

  

  

  We'll follow the same basic idea to find the other solution. Let's reset by letting A=%28matrix%282%2C2%2C5%2C2%2C-2%2C6%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%2C5%2Chighlight%2821%29%2C-2%2Chighlight%28-34%29%29%29

  

  

  

  Now compute the determinant of A%5By%5D to get abs%28A%5By%5D%29=%285%29%28-34%29-%2821%29%28-2%29=-128.

  

  

  

  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=%28-128%29%2F%2834%29=-64%2F17

  

  

  

  So the second solution is y=-64%2F17

  

  

  

  

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

  

  Final Answer:

  

  

  

  

  So the solutions are x=97%2F17 and y=-64%2F17 giving the ordered pair (97/17, -64/17)

  

  

  

  

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