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

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. 
x + 2y + 3z = 4 
4x + 3y + 2z = 1 
-x - 2y - 3z = 0

Accepted Answer

Hello, everyone. We are asked to solve the system of equations by using Kramer's rule and use a different method. If D equals zero, the system of equations we are given has three equations negative X plus five, Y plus eight, Z equals nine seven, X minus 10, Y plus three, Z equals two, X minus five, Y minus eight Z equals four. Our answer choices are a, the set negative 262 B, the set 51 negative four C since D equals zero, Kramer's rule does not apply. And we get the answer in parentheses. Five Z minus one divided by seven in parentheses, negative two Z plus five closed parentheses divided by three Z or D since D equals zero, Kramer's rule does not apply. And our answer is the null set. First recall that when you're doing Kramer's rule to solve a system of equations, we must first set up a matrix using the coefficients of the variables. Our first equation will be our first row and so on. So our first row is negative one, 58. The second row will be seven negative 10, 3. The third row would be one negative five negative eight from there, we must find the determinant of this to find out what D equals. So recall that the determinant is gonna be the first element from the first row multiplied by the minor matrix comprised of the elements of the rows and columns that do not include in this case negative one. So we have negative one multiplied by a two by two matrix with negative 10 3 as the top row, negative five, negative eight is our second row. Then we subtract the second element in the first row. So five multiplied by the determinant of the minor two by two matrix again, created from the rows and columns that do not include that element five. So they'll give us a top row of 73 in the second row of one negative eight. And then we add the third element eight multiplied by the determinant of the minor matrix. That is again gonna be created by the rows and columns that do not include eight. So the top row would be seven, negative 10 and the second row would be one negative five. From here. I'm going to find the determinant of each of these minor matrices. So I have negative one multiplied by the parentheses. And to recall that to find a determinant of a two by two matrix, we start at the top left corner, we find the product diagonally and then subtract the other diagonal product. So negative 10 multiplied by negative eight would be a positive 80 minus three multiplied by negative five, which is negative 15, which is like 80 plus 15 in parentheses. Next minus five opening parentheses. Seven multiplied by negative eight is negative minus three, multiplied by one which is three closed parentheses plus eight multiplied by the parentheses. And again, diagonally seven multiplied by negative five is negative minus negative 10 multiplied by one which is negative 10, which is like adding 10, close parentheses almost there. So we have negative one multiplied by the parentheses. 80 plus 15, 80 plus 15 is 95. So negative one multiplied by 95 is negative 95. Next, I have negative five multiplied by the parentheses, negative 56 minus three. So in the parentheses, I get negative 59 and negative 59 multiplied by negative five is a positive 295. So plus 2 95 and then plus eight multiplied by the parentheses. Negative 35 plus 10, negative 35 plus 10 is negative 25 negative 25 multiplied by eight is a negative 200. So when we add this all together D equals zero, this means we cannot solve this with Kramer's rule. So we need to use another method. So we know our answer choices are either gonna be C or D. So the method I'm gonna use is gonna be elimination and looking at the original equations. I notice that equation one and equation three look like they could eliminate some variables on their own. So that's what I'm gonna start with. So rewriting that first equation which was negative X plus five Y plus eight Z equals nine. And I'm gonna add that to the equation X minus five, Y minus eight Z equals four. I notice I already have opposite coefficients. So I don't have to multiply either equation by anything. And when I add straight down negative X and X will cancel out five Y and negative five Y will cancel out eight Z and negative eight Z will cancel out. So on the left hand side, I'm left with zero. However, on the right hand side, nine plus four is 13, this is a false statement and being that it is a false statement. This means that there's no solution for our system. So our answer is the empty set. So this means our answer choice has to be D since D equals zero. Kramer's rule does not apply. But the answer to the system of equations is the empty set. 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. x + 2y + 3z = 4 4x + 3y + 2z = 1 -x - 2y - 3z = 0

Key Concepts

Cramer's Rule

Determinants

Alternative Methods for Solving Systems

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