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 methodto determine the solution set. See Examples 5–7. 3x + 2y = 4 6x + 4y = 8

Accepted Answer

Hello, everyone. We are asked to solve the system of equations by using Kramer's rule and use a different method to solve if D equals zero. Our system of equations has two equations negative four X plus seven, Y equals 11 and negative eight X plus 14, Y equals 22. Our answer choices are a B negative 14 C since D equals zero, Kramer's rule does not apply. And then we have the value negative open parentheses 11 minus seven Y closed parentheses divided by four. And why D since D equals zero, Kramer's rule does not apply. And we find that the solution is the empty set. First, we need to find this D that we keep talking about. And that is the discriminate of the matrix that is created by the coefficients of the variables. So the top row of our matrix is negative 47. The second row is negative 8, 14. And recall that the discriminate of A two by two matrix comes from the diagonal products subtracted from each other starting in the top left. So negative four multiplied by 14 minus seven, multiplied by negative eight, negative four, multiplied by 14 is negative 56. And I'm gonna treat this as negative seven multiplied by negative eight, which is a positive 56. Well, 50 negative 56 positive 56 equals zero. So D happens equal zero, which means Kramer's rule does not apply here. So we need to solve this using a different method. Typically when I solve systems of equations, I use either substitution or elimination. Since a, a quick glance, I noticed I could multiply the top equation by negative two and cancel out my X values. I'm gonna use elimination. So four X plus seven Y equals 11 and negative eight X plus 14 Y equals 22. Where again our two original equations using elimination, I'm gonna multiply the top equation by negative two and I get eight X minus 14 Y equals negative 22. My bottom equation does not change negative eight X plus 14 Y equals 22. When I add them together and combine my like terms eight X and negative eight XX cancel out, which is what I wanted. Negative 14 Y and 14 Y also cancel out as do negative 22 22. So this means we get zero equals zero. But the good news is this is a true statement which means there are infinitely many solutions. However, we need to now figure out what, how we're gonna write those solutions. Well, we can rewrite them in terms of Y meaning instead of having X equals and Y equals, we're gonna find out what X equals. When we start with the equation negative four, X plus seven, Y equals 11. We could use the other equation also, but this one has smaller numbers and we want to solve for X. So we're gonna get the X value in terms of Y. So solving for X, I'm gonna subtract seven Y from both sides. Do I have negative four, X equals negative seven Y plus 11, dividing both sides by negative four, I currently have X equals negative seven Y plus over negative four. I want to work with that negative sign. Um So I am going to rewrite my numerator. So I don't have a leading negative so I could have 11 minus seven Y and I'm gonna put that in parentheses. And if you recall, you can bring the negative from a denominator to a numerator in a fraction without changing the value. So this means in our solution, the X value is negative parentheses 11 minus seven Y over four. And since we're writing this in terms of why, why is gonna equal Y? So our solution is the set negative 11 minus seven Y over four comma Y and that is the solution we are given in answer choice C that Kramer's rule does not apply and that negative and then parentheses 11 minus seven Y over four and Y are the solution? Have a nice day.

Use Cramer's rule to solve each system of equations. If D = 0, then use another methodto determine the solution set. See Examples 5–7. 3x + 2y = 4 6x + 4y = 8

Key Concepts

Cramer's Rule

Determinants

Alternative Methods for Solving Systems

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