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Algebra II Textbook26 chapters | 256 lessons

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Lesson Transcript

Instructor:
*Yuanxin (Amy) Yang Alcocer*

Amy has a master's degree in secondary education and has taught math at a public charter high school.

Watch this video lesson to learn how you can easily find the solution of a linear system in two variables by just using determinants. All you need to find the complete solution is to find three determinants.

**Linear systems in two variables** are common in math. You will see them more and more as you progress. They are collections of two equations with two variables and no exponents. There are several methods that you can use to solve these systems.

We will talk about the method that uses matrices and determinants in this video lesson. Take a moment to refresh your memory, if you need, on how to create matrices from equations and also how to find the determinants of matrices.

What kind of a problem might you expect to solve? For example, linear systems in two variables can solve such a problem as how many dogs and cats there are at a pet store.

Say a pet store owner tells you he knows that he has a total of 50 dogs and cats, and he knows that when he subtracts the number of cats from dogs he gets 30. Well, you can totally help him figure out how many cats and dogs he has by creating this linear system in two variables:

The first equation tells you that when you add the number of cats and dogs together you get a total of 50. The second equation tells you that when you subtract the number of cats from dogs you get 30. This is your system.

To help you solve this system, we can use Cramer's rule. **Cramer's rule** tells you how you can solve the linear system by working with just determinants. First, you create your coefficient matrix, which includes only the coefficients on the left side of the equations. Then you find the determinant of this matrix.

We will call this determinant *D*. Then for each variable, you take your coefficient matrix and substitute the constant numbers, which are the numbers on the right side of the equation, into each column.

So, for the *c* variable, you substitute the constant numbers into the first column of the coefficient matrix. For the *d* variable, you substitute the constant numbers into the second column of the coefficient matrix.

You then find the determinants of these matrices. For the *c* variable, we will call the determinant *D* sub *c*. For the *d* variable, we will call the determinant *D* sub *d*.

The solution then is found by division. The *c* solution is *c* = *D* sub *c* / *D*. The *d* solution is *d* = *D* sub *d* / *D*. Because we have two variables, we need to find three determinants: one general determinant and then one determinant for each variable.

Let's look at it in action now. We first create our coefficient matrix:

Finding the determinant, we get 1 * 1 - 1 * -1 = 1 + 1 = 2. So, *D* = 2. This is our general determinant. We now need two more determinants, one for each variable. Next, for the *c* variable, we substitute the first column with the constant numbers of 50 and 30. We get this for the matrix:

Taking the determinant, we get 50 * 1 - 1 * 30 = 50 - 30 = 20. So, *D* sub *c* is 20. Next, we substitute the second column with the constant numbers of 50 and 30 to find *D* sub *d*. We get this for the matrix:

Taking the determinant of this matrix, we get 1 * 30 - 50 * -1 = 30 + 50 = 80. So, *D* sub *d* is 80. We now have three determinants that we need to solve our system. So, remember, if you have two variables, you need three determinants: one general and then one for each variable.

Now, all we have to do to find our answer is to divide our determinants. The *c* solution is *c* = *D* sub *c* / *D* = 20 / 2 = 10. The *d* solution is *d* = *D* sub *d* / *D* = 80 / 2 = 40.

So this means that we have 10 cats and 40 dogs in the pet store. We have successfully solved the problem, and we can tell the pet store owner what we found.

Let's review what we've learned. We learned that **linear systems in two variables** are common in math. They are collections of two equations with two variables and no exponents. You can use Cramer's rule to help you solve such a system.

To use **Cramer's rule**, you first find the determinant of your coefficient matrix, *D*. Then for each variable, you swap out the appropriate column of the coefficient matrix with the constant numbers.

Then you find the determinant of these matrices. You will get *D* sub *x* and *D* sub *y*, one for each variable. The *x* solution is then *D* sub *x* / *D* and the *y* solution is *D* sub *y* / *D*.

After this lesson is done you should be able to:

- Recall Cramer's rule
- Identify and solve a linear system in two variables

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Algebra II Textbook26 chapters | 256 lessons

- What is a Matrix? 5:39
- How to Write an Augmented Matrix for a Linear System 4:21
- How to Perform Matrix Row Operations 5:08
- Matrix Notation, Equal Matrices & Math Operations with Matrices 6:52
- How to Solve Inverse Matrices 6:29
- How to Solve Linear Systems Using Gaussian Elimination 6:10
- How to Solve Linear Systems Using Gauss-Jordan Elimination 5:00
- Inconsistent and Dependent Systems: Using Gaussian Elimination 6:43
- Multiplicative Inverses of Matrices and Matrix Equations 4:31
- How to Take a Determinant of a Matrix 7:02
- Solving Systems of Linear Equations in Two Variables Using Determinants 4:54
- Using Cramer's Rule with Inconsistent and Dependent Systems 4:05
- How to Evaluate Higher-Order Determinants in Algebra 7:59
- Go to Algebra II: Matrices and Determinants

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