Having covered how to manipulate and evaluate determinants, now we'll explore one of the practical uses of determinants, which is in solving systems of equations. Consider the following system of equations:

2x + 3y = 22
3x - 2y = 7

We could solve this system of equations the old-fashioned way, but we can also do it using determinants. Let's suppose that we have been asked to find the value of y in this system. The first thing we do is we create a determinant out of the coefficients on the left-hand side. I've named this determinant d, because we're going to use it as a denominator:

d = = 2(-2) - (3)(3) = -13

Now I'm going to create another determinant by replacing the coefficients of y with the values on the right-hand side of the equation. I'm going to call this n_y, because we're going to use it as a numerator to help us find y.

n_y = = 2(7) - 22(3) = -52

Now, to find the value of y, we just calcluate the following: = = 4.

If we want x, we need to calculate n_x = = 22(-2) - 3(7) = -65.

Thus, x = = = 5.

Our solution is (5,4)

Systems with Three Unknowns
This process may actually be more work than the "old-fashioned" method, if you're solving a system of two equations in two unknowns, but if you have three unknowns, it becomes a bit more useful. Especially if you only need to know one of the three unknowns.

Here's an example. Solve for y in the following system:

2x + 5y + 3z = 47
x - 2y + 5z = 38
3x + y + z = 23

d = = 77

Since we only need to find y, we just need n_y:

n_y = = 231

y = = = 3

Note that we always need to make sure we have the equations formatted the same way (variables in the same order, with the constant on the opposite side of the equation). Also note that if an equation is missing a variable, we need to include it anyway. For example, if you have a system of equations in x,y, and z, and one of the equations is x + 3z = 5, it needs to be written as x + 0y + 3z = 5.

But Wait...
Not all systems of equations have a solution, right? If two of the linear equations are parallel, there is no solution, and if two of the linear equations are equivalent, then there could be an infinite number of solutions! So how does Cramer's Rule trap this issue?

It's actually pretty simple. If two of the equations are parallel or identical, then their coefficients are either equal or multiples of the other, right? And they become rows in a determinant, and all we need to do is subtract the multiple times the row of smaller coefficients times the row of larger coefficients, and we have a row of zeroes, making the determinant zero.

In other words, whenever the system of equations is indeterminate or inconsistent, Cramer's Rule gives you a division by zero, which tells you you can't get solutions. Pretty nice, huh?

1.

Find x if x + 5y = 22 and 10x - y = 67

2.

Find x if x + 2y = 10 and 3x + y = 5

3.

Find z if x + y + z = 8; x - 2y + z = 11; and 2x + y - 3z = -23

4.

Find y in problem #3

5.

Find x,y and z if x + y = 11; y + z = 6; and x + z = 9