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Cramer's Rule - Problem 1

Teacher/Instructor Norm Prokup
Norm Prokup

Cornell University
PhD. in Mathematics

Norm was 4th at the 2004 USA Weightlifting Nationals! He still trains and competes occasionally, despite his busy schedule.

Let’s use determinants to solve a system. We have this system here; 9x minus 6y equals 18, 6x minus 4y equals 24.

The first step is just to recall Cramer’s Rule. We have on the bottom, the determinant of the coefficient matrix, 9 -6, 6 -4. And in the numerator, you want the same determinant, only, in place of the x coefficients, you want these numbers; 18 and 24. So I’ll put the y coefficients there, and I’ll put down the numbers 18 and 24 right here, in the x position. Let’s calculate the denominator first.

We’re going to have 9 times -4, -36, minus 6 times -6, minus -36 plus 36. We see that we actually have a problem here. We have 18 -6, 24 -4 on top and we’ve got zero on the bottom. So we can’t solve this system using Crammer’s rule. I want to figure out what the problem is here.

What does that mean, that you can’t solve this system? What does it mean that the denominator is zero? What does it mean that the determinant of the coefficient matrix is zero? Let’s take a look at the graphs. These two lines are parallel.

Now when you’re solving a system, you’re looking for the intersection of two lines. And if they’re parallel they never intersect. So it makes sense that Cramer’s Rule or any method that you use isn’t going to find a solution. But in this case, when you have two lines that are parallel, the determinant of their coefficient matrix, 9 -6, 6 -4, that determinant is zero. That will always happen when you have two parallel lines.

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