Use Cramer's rule to solve each system of equations. If D = 0,...

Question

Use Cramer's rule to solve each system of equations. If D = 0, then use another method to determine the solution set. See Examples 5–7.

2x - y + 4z = -2

3x + 2y - z = -3

x + 4y - 2z = 17

Accepted Answer

Hello, everyone. We are asked to solve the system of equations by using Kramer's rule. If D equals zero, we are asked to use a different method to solve the system. The system we are given is three equations negative X plus two, Y plus five, Z equals 39 four, X minus eight, Y plus six, Z equals negative 52 nine, X plus Y plus three, Z equals negative 26. Our answer choices are a, the set negative 754 B, the set negative 47, negative five. See the set negative or D since D equals zero, Kramer's rule does not apply. And the answer is the empty set. The first thing to recall is that with Kramer's rule, we are going to set up a three by three matrix that is comprised of the coefficients of the variables. So this would give us row one is negative 125, row two is four, negative 86, row three is 913. We want to find the determinant of this three by three matrix. Recall there are numerous steps to this. And if you need to review them, please go back to the video on how to find the determinant of a three by three matrix and review it before you get too deep into this video for this three by three matrix. We'll end up with the multiplication negative one multiplied by negative 24 minus six, closed parentheses minus two, multiplied by 12 minus 54 closed parentheses plus five, multiplied by four plus 72 closed parentheses. Simplifying all of this, I get 30 plus plus 380 and that equals 494. And that's D so D is not zero. So we can continue with Kramer's rule. Next, recalling Kremer's rule, we are going to find the determinant of three more matrices. The first one being D of X which will be comprised of the first column is the answers from our system. So 39 negative 52 negative and then the other two columns are the same. So we're only replacing the X column with the solutions. So going down to negative 81 and then next column 563, finding the determinant of this three by three matrix. We have 39 multiplied by negative 24 minus six closed parentheses minus two multiplied by negative 156 plus 156 closed parentheses plus five multiplied by negative 52 minus 208 closed parentheses. So this simplifies to 39 multiplied by negative 30 minus two multiplied by zero. Plus five multiplied by negative 260. And so we have negative minus 1300 because the zero will cancel out and this equals negative 2470. So D FX is the value negative 2470. Next, we need to find D of Y. So this is gonna be the determinant when we replace the column, the second column, which would be the Y coefficients with the solutions. I'm gonna put that right here. So D F Y we have negative 149 is our first column. 39 negative 52 negative 26 is our second column 563 as our third column. And again, we are gonna find the determinant of this. And when we do so we have negative one multiplied by negative 156 plus 156 minus 39 multiplied by the parentheses, 12 minus 54 closed parentheses plus five, multiplied by the parentheses, negative 104 plus 468. Simplifying my parentheses, I get negative one multiplied by zero minus 39 multiplied by negative plus five, multiplied by 364. And this works out to be zero plus 106 or sorry, 1638 plus 1820. And in the end, we have 3458. So that's D FX 58 one more matrix C to solve. And then we can find our solutions. So we need to find D of Z. I'm gonna put this right here because this is the biggest open space I have. So in this one, we're going to replace the third column with our solution column from our system. So we have going down in columns negative 149, then two negative 81 and then 39 negative 52 negative 26. And what this looks like we have negative one multiplied by 208. And actually I'm gonna move that down here just so we have a little bit more space. So negative one multiplied by 208 plus 52 closed parentheses minus two multiplied by negative plus and then plus 39 multiplied by, and I'm gonna add them together to save space 76. So, simplifying this, I get negative one multiplied by 260 minus two, multiplied by and then plus 39 multiplied by 76. And this will result in negative minus plus 2964. And that totals to a positive 1976. So D FX is 1976. All right, we've done all the heavy lifting almost there to find the final values. Here. We are gonna take the effects and divide it by D. So we'll get negative 2470 divided by 494 which divides to negative five. Next D of Y divided by D D F Y was 3458 divided by 494 which is seven and D of Z divided by D. So 1976 divided by 494 and that will equal four. So this means that the X value is five, the Y value is seven and the Z value is four. So the solution set is answer choice, C negative have a nice day.

Use Cramer's rule to solve each system of equations. If D = 0, then use another method to determine the solution set. See Examples 5–7.
2x - y + 4z = -2
3x + 2y - z = -3
x + 4y - 2z = 17

Key Concepts

Cramer's Rule

Determinant of a Matrix

Alternative Methods for Solving Systems

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