In Exercises 45–48, explain why the system of equations cannot be...

Question

In Exercises 45–48, explain why the system of equations cannot be solved using Cramer's Rule. Then use Gaussian elimination to solve the system. 4x - 3y - 2z = 128x - 6y - 4z = 22

Two equations from a system for Exercise 47 in college algebra on determinants and Cramer's Rule.

Accepted Answer

hey everyone here we are asked to use Gaussian elimination to solve the system. By first explaining why we cannot use Cramer's rule. Well first our given equations are x minus four, Y plus seven, Z is equal to negative 11 negative five, X plus 20 Y minus 35 Z is equal to 27. And so we first need to take a look at these equations here and note that we have three variables, but only two equations. So we will have a non square matrix. And because our matrix is non square, we cannot find its determinant. And therefore we can not use Cramer's rule to solve the system. So from here we can move on to using Gaussian elimination. So the first step is to set up these two equations into an augmented matrix. So on the left hand side we have the coefficients for the variables. And then we place this bar to represent the equal sign from the equations and then the right hand side is simply the right hand terms. So for the first row we have from our first equation, one negative 47 and then negative 11. And taking a look at our second equation. For the second row we have negative 5, 20 negative 35 27. So now we need to perform Gauss Jordan elimination. So to do this we can first focus on getting this bottom left term, this negative five into a zero. So the first thing I notice is that the first column here has a one and a negative five. Well we can multiply this one by five and then add it to negative five and we get zero. So with this information we take the first row and multiply all of these terms by five. So we have five times row one and then we add it to row two, so plus row two. So with this our matrix is for the first row. We multiply it all by five. So we have five negative and then equal to negative 55 our second row is the same. So we have negative 5, 20 negative 35 27. Now just adding the two rows together, we get a result here on the left hand side we have five plus negative five which is zero, negative 20 plus 20 is just zero and 35 plus negative 35 is also just zero. And then on the right hand side negative 55 plus 27 is negative 28. And so now we take this result and we use it to replace this second row from our original matrix. So now our manipulated matrix is for row one. The same one negative 47 is equal to negative 11. And our second row is now 000 negative 28. Well, with this bottom row here it's the same thing as saying zero x plus zero, y plus zero, Z is equal to negative 28 which is the same thing as saying zero is equal to negative 28 which we know is untrue, and therefore we know that this system here has no solution. So in summary, we have a non square matrix. Therefore, we cannot use Cramer's rule, and we have found using Gaussian elimination, that our system has no solution. Therefore, we're left with answer choice. D. Thanks for watching. I hope you found this video helpful, and I'll see you again next time.

In Exercises 45–48, explain why the system of equations cannot be solved using Cramer's Rule. Then use Gaussian elimination to solve the system. 4x - 3y - 2z = 128x - 6y - 4z = 22

Key Concepts

Cramer's Rule

Gaussian Elimination

Dependent and Inconsistent Systems

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