Tutoring high school math, I don’t see this used.  However, I did see it at university.  The tutor introduces Cramer’s Rule.

Consider the following problem, common in high school math:

Solve the system.

2x-4y=20
3x+5y=-3

If you know determinants (having read my articles here and here), then you have the option to use Cramer’s Rule.

Cramer’s Rule sees the problem as follows:

The original system, in matrix form, is called the augmented matrix – so-called because it contains the constant numbers in the third column:

The matrix with just the x and y columns is called “the 2×2 matrix”:

With Cramer’s Rule, we evaluate determinants of various matrices derived from the augmented matrix. The solution to each variable begins in fraction form. Its numerator is a 2×2 determinant in which the third column from the augmented matrix is substituted for the column of the variable being found. The denominator is the determinant of the 2×2 matrix. Let’s proceed:

Solving the above system for x, Cramer’s Rule gives

Notice that, in the numerator, the third column from the augmented matrix is substituted for the $x$ column, while the $y$ column stays the same.

Then, Cramer’s Rule gives, for y:

We evaluate the determinants in each case, then simplify. First for x:

x=(100-12)/(10+12)=88/22=4

We do the same for y:

y=(-6-60)/(10+12)=-66/22=-3

Apparently, x=4,y=-3. Let’s sub the values back into our original equations just to make sure:

2(4)-4(-3)=20 correct.

3(4)+5(-3)=-3 correct.

Since, in each of our original equations, the left side equals the right side, we have confirmation that our solution x=4,y=-3 is correct.

Cramer’s Rule can be very handy, especially for 2-variable systems.

I’ll be saying more about Cramer’s Rule and systems of equations in future posts:)

Source:

Johnson/Riess/Arnold. Introduction to Linear Algebra, 2nd edition. Don Mills: Addison-Wesley, 1989.

Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.