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. 
1.5x + 3y = 5
 2x + 4y = 3

Accepted Answer

Hello, today we're going to be solving the system of equations by using Kramer's rule. Now, the first thing to do in order to use Kramer's rule is we need to convert the system of equations into a system of matrices. So what we're going to do is we're going to create a two by two matrix that's going to hold the coefficients of all of our X and Y terms. Then we're gonna create a matrix that's going to hold the solutions to each of the rows. So for row one row, one is going to hold the coefficients of the X and Y turns, that's gonna be negative 2.5 and four. And the solution to the first row is going to be seven next the second row or at least the second row is going to have the coefficient of X and Y which is going to be negative five and eight. And the solution to the second row is going to be three. Now, we need to verify that we can use Kramer's rule in order to do that, we're going to take the determinant of the two by two matrix that we just created that is gonna be negative 2. negative five and eight. Now, in order to take the determinant, we're going to multiply the upper left element with the lower right element and subtract the product of the lower left and upper right. The determinant for this two by two is going to be negative 2. times eight minus negative five times four, negative 2.5 times eight is going to give us negative and negative five times four is going to give us negative 20 times negative one is going to give us positive 20. A negative 20 plus 20 is going to give us zero. Now because the determinant is equal to zero. That means we cannot apply Kramer's rule. We have to use a different method in solving the system of equations. Well, we can go ahead and solve the system of equations by using the method of graphing. So if we take a look at the first equation, we have negative 2.5 X plus four Y is equal to seven. This is the equation in the standard form of the equation of a line. So if we solve for Y, we can get the point intercept form of the equation of the line. We're going to add 2.5 X to both sides of the equation to sol for Y. And that is going to give us four, Y is equal to 2.5 X plus seven. And if we divide everything by four, we get Y is equal to 0. X plus 1.75. So that's going to be the equation of the first line. Now, if we take a look at the second equation, we get negative five X plus eight Y is equal to three. We're gonna convert this into the point intercept form of the equation of the line by solving for Y. So we're gonna add five X to both sides of the equation leaving us with eight Y is equal to five X plus three. And if we divide everything by one by eight, we get Y is equal to 0. plus 0.375. Now, one thing to note is that the slopes for both of these lines are exactly the same and that means that these lines are going to be parallel lines. So just to draw a rough graph, the second equation is going to start at 0.375 and have a positive slope. And the first equation is going to start just slightly above this line. But since they have the same slope, they're going to be parallel lines and they're never going to intersect. And since there is no intersecting point between the two lines, there's going to be no solution to the system of equations. And with that being said, the answer to this problem is going to be D so I hope this video helps you in understanding how to approach this problem. And I'll go ahead and see you all in the next video.

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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. 1.5x + 3y = 5 2x + 4y = 3