Solve the system of equations using Cramer's Rule.4x+2y+3z=64x+2y+3z=64x+2y+3z=6x+y+z=3x+y+z=3x+y+z=35x+y+2z=55x+y+2z=55x+y+2z=5

x = 1 , y = 4 , z = − 2 x=1,y=4,z=-2 x = 1 , y = 4 , z = − 2

Solve the system of equations using Cramer's Rule . 4 x + 2 y + 3 z = 6 4x+2y+3z=6 4 x + 2 y + 3 z = 6 x + y + z = 3 x+y+z=3 x + y + z = 3 5 x + y + 2 z = 5 5x+y+2z=5 5 x + y + 2 z = 5

Welcome back, everyone. I'm just gonna jump right into this practice problem because we're gonna use Cramer's rule for a system of 3 equations with 3 variables to solve this system of equations over here. Now remember, Cramer's rule for these types of problems is very, very tedious. There's lots of calculations of determinants, so we're just gonna jump right in. Alright? So what I have to do is I'm gonna take the system of equations, and the first thing I have to do is actually convert it into a matrix because I need to figure out what all my columns and numbers are, a, b, and c, and d. Alright? So I'm gonna convert this into a matrix. Remember, all I'll have to do here is just pull out the coefficient. So this is gonna be 4, 2, 3. The black bar means an equal sign. This is gonna be 1, 1, 1, and 3, and this is gonna be 5, 1, 2, and 5. Alright? So now that I've turned this into a matrix here, remember that I've got, columns for the coefficients. Right? This is gonna be my a column for the x coefficients, b for the y coefficients, and so on and so forth. So it'd be c. And then finally, I've got the slower case d column over here. Alright? So remember what happens is that Cramer's rule just directly will give us the solutions for x, y, and z. These are the numbers that will plug into these equations and make them true. And, really, we have to solve this by calculating a bunch of 3 by 3 determinants. It's actually a quotient of determinants, and what you have to do is notice that all of these things are divided by this, capital d over here, which is really just the 3 by 3 matrix that you get by looking at all the coefficients of x, y, and z. So that's always a good place to start looking at this capital d matrix because you have to divide by that number in all of your equations over here. Alright? So let's get started here. The first thing you want to do is actually want to calculate what that capital, d is. Alright? So really, what it is is you're going to take the matrix of just the coefficients. So in other words, this is just gonna be the first three columns. So this is gonna be 423111, and this is gonna be 512. Alright? So you're going to calculate the determinants of this matrix. And just as a refresher, remember how determinants work. You're going to take a look at all the first entries or the first numbers in that first row, and you're gonna have to separate them into and break them down to smaller matrices. So, for example, the first term is gonna be 4 times the smaller matrix over here. It's gonna be 1 one one two, then you're gonna minus this is gonna be this number times this smaller matrix over here. So in other words, 2 times, 11552, and then this is gonna be 3 times this smaller matrix, over here. So this is gonna be 1151. Alright. So just gonna work this out really quickly here. Notice how a lot of these are 1, so it makes the math a little bit easier. So let's take a look. Right? This is gonna be 4. Now this just says 1 times 2 minus 1 times 1. So in other words, this is 1 minus 2. I'm sorry. This is gonna be 2 minus 1, and so, therefore, you're just gonna get 4 times 1. Alright? Let's take a look at the next one. So this is 1 times 2, so it's 2, minus 1 times 5. So in other words, it's just 2 minus 5. So if you work this, out, what you're gonna get is this is just negative 3. Alright? And if you look at this over here, this is gonna be 1 times 1 minus 5 times 1 or 1 times 5. In other words, that's just gonna be 1 minus 5, and that works out to negative 4. Alright? So we kind of, you know, just did some some quick math there just because a lot of them turned out to be 1, so it makes math a little bit easier. So let's take a look. Right? So this just becomes 4. We have minus 2 times negative 3. The minuses will cancel, and you'll end up with plus 6. Then we have 3 times negative 4, which is gonna be minus 12. You work all that out. 4+6 is 10, minus 12, so that you'll see that your capital d is equal to negative 2. Alright? So, remember, that's not our x, y, and z numbers, but we'll need this negative 2 for all of those calculations involving x, y, and z. Alright. So we kind of have to figure this out. But now we can get into the fun part, which is actually figuring out what x, y, and z are. So let's go ahead and do that. Now remember, what happens is for x, you're going to divide this thing, this new determinant called dx divided by the capital d that you just calculated. So in other words, it's just gonna be some number divided by capital d, which is negative 2. And once we figure that out, that's actually going to be one of our special numbers. That's actually going to be one of our numbers that we're really looking for here. Alright? Now to get that, we have to figure out, well, what is dx? And remember, the idea here is that you're basically just going to take one of the columns, the coefficients of your x columns, and you're gonna replace them with the constants on the right side of the equation. You replace the d column with the a column. Alright? So really what happens is it's just gonna be this 3 by 3 matrix. All of these numbers in the y and z columns will stay the same. So in other words, it's gonna be 211, 312. But now instead of 415, we're actually just gonna use these, these numbers that are on the right side of the equal sign. So this is going to be 6, 3, and 5. Alright? So now we just have to calculate the determinant of this matrix. So I'm just gonna do the same exact thing I did over here. Alright? So remember, this is just gonna be 6 times the smaller matrix. This is gonna be 1, 1, 1, 2 minus 2, and this is gonna be times the smaller matrix. This is gonna be 3512, and this is gonna be plus 3. And then the smaller matrix over here is gonna be this 1. So it's gonna be 3511. Alright? So let's work at each one of these things, independently or sort of 1 by 1. So this is just gonna be equal 6. And then this is gonna be, well, 1 times 2. We actually already figured out what this matrix ends up being. If you look over here, we already had what that matrix is, and it just works out to just be 1. So a lot of times what's going to happen here is you'll end up with the same 2 by 2 determinants throughout the problem. So if you've already sort of you know, if if one seems familiar to you, it's probably because you already probably calculated it before. Alright. So this is gonna be minus 2. Now this one is one we haven't seen yet, so we're gonna have to calculate this as as 3 times 2, which is 6, minus 1 times 5, which is 5. So in other words, this is just 6 minus 5, and this works out to just be 1. Alright? And then we have plus 3. This is gonna be 3, times 1 minus 5 times 1. So, in other words, it's just 3 minus 5, which just turns out to be minus 2. Alright? So let's, just be careful here because there's lots of, you know, negative signs that could pop up throughout your problem. So just be careful. Alright. So this works out to just be 6 minus, 2 times 1, which is 2, and then plus 3 times negative 2, which is minus 6. So one one of the things you'll actually notice is that the 6 and negative 6 will cancel, and what you're left with is actually just that d x is equal to negative 2. Alright? So remember, d x actually goes up here in the numerator of this equation. X is equal to d x over d. So in other words, it's just equal to negative 2 over negative 2, and that just turns out to be 1. So that's one of our magic numbers. So we're 33% of the way done with this problem. Now we just have to do the exact same thing, sort of rinse and repeat for the y and z. Alright? So we're just gonna go a little bit faster because we're actually gonna see that we've already solved some of the determinants that we'll be looking at later on. So this is gonna be y equals dy over d. So same thing. We're going to have to take some number and divide it by negative 2. And once we figure that out, that's going to be one of our second magic numbers. All right? So how do we get the dy? Well, in over here for the x, we replace the constants, the low the capital or sorry, the lowercase d with the x column. Now we're just going to do it for the y column. Alright? So what happens is this dy over here is going to be the determinants of the smaller matrix over here, and we're going to take a look up here at our equation. So now we're going to have is we're going to have the 6, 3, and 5 replacing this column over here. So, in other words, the x and z columns are going to be the same. So what we see here was that we have 415, and then we have 635, and then we have 312. Alright? So what is this matrix, or what is the determinant? It's gonna be 4 times the smaller matrix. This is gonna be 3152. And if you look at this, we've already actually figured out what that determinant is, so it's gonna help us later on. And minus we're gonna do 6 times the smaller matrix over here. This is gonna be 15, and then we're gonna have 12 plus 3, and this is gonna be 1, 3, and then 55. Alright? So let's take a look. We've actually already figured out what 2 of these matrices are. We've seen this one, and we've seen this one. So in other words, this is just we've already figured this out. This actually just works out to just be 1, because that was this one over here. This 1152, this one over here, we've already seen that. And when we calculated when we calculated this one over here, this also works out to so this works out to be negative 3, but you can always just solve that on your own and just double check. So this is just going to be minus 6 times negative 3, and this is going to be 3, and this is going to be 1 times 5. So this is going to be 5 minuteus 3 times 5. This is gonna be 5 minus 15. Alright? So this actually works out to be negative 10. We actually haven't calculated that one yet, but you know? So that's what that is. Okay. So in other words, this is just gonna be 4 minus well, actually, we're gonna have plus 18 because these minuses will cancel. And then 3 times negative 10 will actually be subtracting 30. So in other words, this is just gonna be 22 minus 30, and you'll end up with dy is equal to negative 8. So now we plug this into our numerator. That's dy. So negative 8 divided by negative 2 is equal to 4. Alright? So that's the second magic number. Alright? We're 66% of the way done with this problem now. Homestretch or almost done. Now we just have to figure out z. So z is just gonna be similar. We're gonna do d z over d. In In other words, some number divided by negative 2, and that's going to be our last magic number. And we get this by looking at the dz and replacing the constants in the z column now. So in other words, this is just going to be the determinant. We're going to have 4, 1, 5, and then this is gonna be, 211 and then 6, 3, 5. Alright? So this is going to end up being 4 times the smaller matrix of 1315 minus 2. Smaller matrix over here is gonna be 1355. We've actually already figured out what this one is. Plus 6, and this is gonna be now 1151. So let's just do this real quickly here. Almost done. So this is just gonna be, this is gonna be 4. This is gonna be 5 minus 3. This is gonna be 2. This is gonna be minus 2. Now this is just gonna be let's see. We have 1 minus 5 minus 3 times 5. We already figured that this out. This is gonna be negative 10. That was this determinant over here. Then we have plus 6. This is gonna be 1 times 1 minus 5 times 1. So, what you'll see here is that this actually ends up being, 1 minus 5, which is negative 4. So in other words, this turns out to be negative 4 over here. Alright? So with it, what you end up getting here is 8. You could end up getting 8 minus. It actually will get 8 plus. This is gonna be 20, and then you have minus 24. So in other words, you get the d z is equal to this would be 28 minus 24, which is gonna be 4. And when you plug this back over here, what you're gonna see is that this is ends up being negative 2. Alright? So in other words, the solution to your system of equations over here is that x is equal to 1, y is equal to 4, and z is equal to negative 2. Again, very tedious way of getting the solutions to a system of equations, but you may need to know how to do this. Hopefully, this helps, and thanks for watching.

Solve the system of equations using Cramer's Rule . 4 x + 2 y + 3 z = 6 4x+2y+3z=6 4 x + 2 y + 3 z = 6 x + y + z = 3 x+y+z=3 x + y + z = 3 5 x + y + 2 z = 5 5x+y+2z=5 5 x + y + 2 z = 5

x = 1 , y = 4 , z = − 2 x=1,y=4,z=-2 x = 1 , y = 4 , z = − 2

x = − 2 , y = − 8 , z = 4 x=-2,y=-8,z=4 x = − 2 , y = − 8 , z = 4

x = 2 , y = 8 , z = − 4 x=2,y=8,z=-4 x = 2 , y = 8 , z = − 4

x = − 1 , y = − 4 , z = 2 x=-1,y=-4,z=2 x = − 1 , y = − 4 , z = 2

18. Systems of Equations and Matrices

Determinants and Cramer's Rule

Precalculus

18. Systems of Equations and Matrices

Determinants and Cramer's Rule

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Multiple Choice

Solve the system of equations using Cramer's Rule.
$4x+2y+3z=6$
$x+y+z=3$ $5x+y+2z=5$

$x=-2,y=-8,z=4$

$x=1,y=4,z=-2$

$x=2,y=8,z=-4$

$x=-1,y=-4,z=2$

Verified step by step guidance

Identify the system of equations: 4x + 2y + 3z = 6, x + y + z = 3, and 5x + y + 2z = 5.

Write the coefficient matrix A, the variable matrix X, and the constant matrix B. A = [[4, 2, 3], [1, 1, 1], [5, 1, 2]], X = [[x], [y], [z]], B = [[6], [3], [5]].

Calculate the determinant of the coefficient matrix A, denoted as det(A).

Create matrices A_x, A_y, and A_z by replacing the respective columns of A with the constant matrix B. For example, A_x is formed by replacing the first column of A with B.

Calculate the determinants of A_x, A_y, and A_z, denoted as det(A_x), det(A_y), and det(A_z). Use Cramer's Rule to find the solutions: x = det(A_x)/det(A), y = det(A_y)/det(A), z = det(A_z)/det(A).

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