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Solve using augmented matrix method: If the system has infinite number of solutions, state the general solution. If the system has no solution, explain why: X1 + 3x...

Question

Solve using augmented matrix method: If the system has infinite number of solutions, state the general solution. If the system has no solution, explain why: X1 + 3x2 = 13x1 2xz = 14

Solve using augmented matrix method: If the system has infinite number of solutions, state the general solution. If the system has no solution, explain why: X1 + 3x2 = 1 3x1 2xz = 14



Answers

Determine the general solution to the linear system $\mathbf{x}^{\prime}=A \mathbf{x}$ for the given matrix $A$. $$\left[\begin{array}{rr} 3 & 13 \\ -1 & -3 \end{array}\right]$$

From the human system of the American visions, You can write the human Terry metrics. As for the coefficient off expert in the first equation is three coefficient of extreme minus two. On the constant is plane, but registration second room. Suppose we send off experiments school, the more efficient off. Explain, explain and the constant ID 3 30 lamented three. Now third room, The coefficient of explaining seven. The coefficient off X two is three in the course. 20. Now look at the second example The coefficient off expert in the firstly in regulation is toe excuse Anything They're left like zero the coefficient off extreme too. On the question is But now let us right the elements off. Second, the coefficient off. Explain is three coefficient off experience minus one. The coefficient off extreme is for and the constant is seven. Now from the Thirdly in deregulation will write the elements off total the coefficient of explain it. Six. The proficient off extra in Israel, the way efficient off experience minus mint. And the constant is, you know, look at the third example. In third example, three questions are human and from all three questions we can see that the system offline your content three variables X one x two extremes in the first ignition. The coefficient off explain is one exposed missing. Therefore with zero in first equation Extreme is missing. There will be zero and at last the constant be right. So the elements in the first through our 1001 Now look at the second equation. X one is the thing Will therefore will write zero the coefficient off experience one extreme missing. They're very will take zero in the constant is what Look at the throat and the last minute exhibition explain is missing. The referee will take zero experience. Nothing there for second eliminated Those zero extreme Is there an ex coefficient? Is one there? But it will take one on the constant is three. This is how from even linear situations We can raise the human Parametric

In this question for the give insert off questions I can make an augmented metrics as 213 one, minus three and 12. And I use the transformation are goes to are two minus are won by two. Okay, so that gives me 13 as artists, this become zero minus three, minus one by two. That gives me minus seven by two. All right, And this will be world minus three by two. So that gives me 21 by two. So this can be contorted. All right, are two goes toe are to indo, minus little by seven. Okay, so that gives me 13 Zito one and this becomes minus three. All right, so from here, I will get any question, lex. Plus, why is equal to three on Why is it called? Oh, my industry. So, using all this, I'll get Y is equal to minus three, as it is very evident on the X value will be often does pretty. So that's my answer.

Core system like this with fractions in it. If you don't want to have to deal with the fractions, you can start by multiplying the first equation by two and the second equation by two. So now we have two x minus three y equals 26 and we have three x minus two. Y equals 34 and now we can proceed with the elimination method. So let's see what we can do to get our coefficients on X or Y to be opposites. So if we go for getting our coefficients on X to be opposites, we could multiply the first equation by three. And we could multiply the second equation by negative, too. That way, the coefficients on X will be opposites. So when we multiply the first equation by three, we have six. X minus nine y equals 78. And when we multiply the second equation by negative two, we have negative six X plus four y equals negative 68. So you see, we have opposite coefficients on X, and so when we add the equations together, the X terms are eliminated and we're left with negative five y equals 10. Divide by negative five. And we have y equals negative, too. Now, let's go back to one of our previous equations to find the value of X. So suppose I use X minus three halves times? Why? Now that we know why is negative? Two equals 13. So that would be X plus three equals 13. So now we know X is 10. So our solution is the 0.10 negative too.

So we have two major cities in an equation, and we have one matrix being multiplied by variable and this rare ball represents and unknown matrix. And this is equal to a another single matrix. So we want to solve for this variable X. And how we're gonna do this is we're gonna, uh, take this matrix. And essentially, as we normally would with any equation, divide this matrix. Now, how do we divide Mitrice's? Well, what we're gonna do? Let's label this one me tricks, eh? And this Matrix B So we can say that X I'm gonna write here X is gonna be equal to matrix be the ride of their natures, eh? And another way we can rent. This is be times the inverse of matrix A. And now this is something that we can solve more easily because we know how to find The Matrix, the inverse of the Matrix using D property to the right. So what we're gonna do first, Let's see if the inverse of a even exists. We're going to take d determine it. Okay, which is gonna be 12 times three, which is 36 minus seven times fired for just 30 father. So this is gonna be one. So now we're gonna solve for the inverse of a and this is gonna be won over. The determinant of a witch is one so one of her one times. And to find the inverse, we're gonna switch out the top left on the bottom. Right. So we're gonna get three and 12 and then we're gonna negate the top right in the bottom left as we have over here. So we're gonna have a negative seven and negative five. And this just turned out to one so we can cross that out. And this is gonna be our inverse matrix. So now all we need to do is multiply be our matrix over here, times the inverse of a So we're gonna do to negative one between two times three 12 they get a seven and negative, and we're going to carry out our normal multiplication process. So we're gonna have the first throw times each column. So we have two times three, which is six plus negative times. Negative five, which becomes positive. Five. Next, we're gonna do two tons making of seven, which is negative. 14 plus negative one times. Negative 12. Negative one times. Positive. 12 Which becomes negative. 12. Next, we're gonna do the second world. We're gonna do three trump story, which is line plus two terms of negative five, which is negative. 10. And we're gonna do the next column. So we have three times naked of seven, which is negative. 21 plus two times swell for just 24. And this gives us a final matrix of 600 plus five is 11 negative. 14 plus negative. 12 becomes negative. 26 nonplussed. Negative 10 because they get if one negative 21 post 24 becomes three. So this is gonna be our matrix. That represents the variable X, huh? Six. Oh, No. Six.


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