# Systems of Equations and Inequalities

## Solving Systems with Cramer's Rule


### Learning Objectives

1. Use Cramer’s Rule to solve systems of equations (IA 4.6.3)

### Objective 1: Use Cramer’s Rule to solve systems of equations (IA 4.6.3)

Cramer’s Rule uses determinants to solve systems of equations.



### Practice Makes Perfect


### Practice Makes Perfect

We have learned how to solve systems of equations in two variables and three variables, and by multiple methods: substitution, addition, Gaussian elimination, using the inverse of a matrix, and graphing. Some of these methods are easier to apply than others and are more appropriate in certain situations. In this section, we will study two more strategies for solving systems of equations.

### Evaluating the Determinant of a 2×2 Matrix

A determinant is a real number that can be very useful in mathematics because it has multiple applications, such as calculating area, volume, and other quantities. Here, we will use determinants to reveal whether a matrix is invertible by using the entries of a square matrix to determine whether there is a solution to the system of equations. Perhaps one of the more interesting applications, however, is their use in cryptography. Secure signals or messages are sometimes sent encoded in a matrix. The data can only be decrypted with an invertible matrix and the determinant. For our purposes, we focus on the determinant as an indication of the invertibility of the matrix. Calculating the determinant of a matrix involves following the specific patterns that are outlined in this section.

### Using Cramer’s Rule to Solve a System of Two Equations in Two Variables

We will now introduce a final method for solving systems of equations that uses determinants. Known as Cramer’s Rule, this technique dates back to the middle of the 18th century and is named for its innovator, the Swiss mathematician Gabriel Cramer (1704-1752), who introduced it in 1750 in Introduction à l'Analyse des lignes Courbes algébriques. Cramer’s Rule is a viable and efficient method for finding solutions to systems with an arbitrary number of unknowns, provided that we have the same number of equations as unknowns.
Cramer’s Rule will give us the unique solution to a system of equations, if it exists. However, if the system has no solution or an infinite number of solutions, this will be indicated by a determinant of zero. To find out if the system is inconsistent or dependent, another method, such as elimination, will have to be used.
To understand Cramer’s Rule, let’s look closely at how we solve systems of linear equations using basic row operations. Consider a system of two equations in two variables.
We eliminate one variable using row operations and solve for the other. Say that we wish to solve for 
  If equation (2) is multiplied by the opposite of the coefficient of 
  in equation (1), equation (1) is multiplied by the coefficient of 
  in equation (2), and we add the two equations, the variable 
  will be eliminated.
Now, solve for 
 

Similarly, to solve for 
  we will eliminate 
 

Solving for 
  gives
Notice that the denominator for both 
  and 
  is the determinant of the coefficient matrix.
We can use these formulas to solve for 
  and 
  but Cramer’s Rule also introduces new notation:
1.  
  determinant of the coefficient matrix
2.  
  determinant of the numerator in the solution of 
  
3.  
  determinant of the numerator in the solution of 
 


The key to Cramer’s Rule is replacing the variable column of interest with the constant column and calculating the determinants. We can then express 
  and 
  as a quotient of two determinants.

### Evaluating the Determinant of a 3 × 3 Matrix

Finding the determinant of a 2×2 matrix is straightforward, but finding the determinant of a 3×3 matrix is more complicated. One method is to augment the 3×3 matrix with a repetition of the first two columns, giving a 3×5 matrix. Then we calculate the sum of the products of entries down each of the three diagonals (upper left to lower right), and subtract the products of entries up each of the three diagonals (lower left to upper right). This is more easily understood with a visual and an example.
Find the determinant of the 3×3 matrix.
1. Augment 
  with the first two columns.
  
2. From upper left to lower right: Multiply the entries down the first diagonal. Add the result to the product of entries down the second diagonal. Add this result to the product of the entries down the third diagonal.
3. From lower left to upper right: Subtract the product of entries up the first diagonal. From this result subtract the product of entries up the second diagonal. From this result, subtract the product of entries up the third diagonal.
The algebra is as follows:

### Using Cramer’s Rule to Solve a System of Three Equations in Three Variables

Now that we can find the determinant of a 3 × 3 matrix, we can apply Cramer’s Rule to solve a system of three equations in three variables. Cramer’s Rule is straightforward, following a pattern consistent with Cramer’s Rule for 2 × 2 matrices. As the order of the matrix increases to 3 × 3, however, there are many more calculations required.
When we calculate the determinant to be zero, Cramer’s Rule gives no indication as to whether the system has no solution or an infinite number of solutions. To find out, we have to perform elimination on the system.
Consider a 3 × 3 system of equations.
where
If we are writing the determinant 
  we replace the 
  column with the constant column. If we are writing the determinant 
  we replace the 
  column with the constant column. If we are writing the determinant 
  we replace the 
  column with the constant column. Always check the answer.

### Understanding Properties of Determinants

There are many properties of determinants. Listed here are some properties that may be helpful in calculating the determinant of a matrix.

### Key Concepts

1. The determinant for 
  is 
  See .
2. Cramer’s Rule replaces a variable column with the constant column. Solutions are 
  See .
3. To find the determinant of a 3×3 matrix, augment with the first two columns. Add the three diagonal entries (upper left to lower right) and subtract the three diagonal entries (lower left to upper right). See .
4. To solve a system of three equations in three variables using Cramer’s Rule, replace a variable column with the constant column for each desired solution: 
  See .
5. Cramer’s Rule is also useful for finding the solution of a system of equations with no solution or infinite solutions. See  and .
6. Certain properties of determinants are useful for solving problems. For example:




### Section Exercises


### Verbal


### Algebraic

For the following exercises, find the determinant.
For the following exercises, solve the system of linear equations using Cramer’s Rule.
For the following exercises, solve the system of linear equations using Cramer’s Rule.

### Technology

For the following exercises, use the determinant function on a graphing utility.

### Real-World Applications

For the following exercises, create a system of linear equations to describe the behavior. Then, calculate the determinant. Will there be a unique solution? If so, find the unique solution.
For the following exercises, create a system of linear equations to describe the behavior. Then, solve the system for all solutions using Cramer’s Rule.
For the following exercises, use this scenario: A health-conscious company decides to make a trail mix out of almonds, dried cranberries, and chocolate-covered cashews. The nutritional information for these items is shown in .

### Review Exercises


### Systems of Linear Equations: Two Variables

For the following exercises, determine whether the ordered pair is a solution to the system of equations.
For the following exercises, use substitution to solve the system of equations.
For the following exercises, use addition to solve the system of equations.
For the following exercises, write a system of equations to solve each problem. Solve the system of equations.

### Systems of Linear Equations: Three Variables

For the following exercises, solve the system of three equations using substitution or addition.
For the following exercises, write a system of equations to solve each problem. Solve the system of equations.

### Systems of Nonlinear Equations and Inequalities: Two Variables

For the following exercises, solve the system of nonlinear equations.
For the following exercises, graph the inequality.
For the following exercises, graph the system of inequalities.

### Partial Fractions

For the following exercises, decompose into partial fractions.

### Matrices and Matrix Operations

For the following exercises, perform the requested operations on the given matrices.

### Solving Systems with Gaussian Elimination

For the following exercises, write the system of linear equations from the augmented matrix. Indicate whether there will be a unique solution.
For the following exercises, write the augmented matrix from the system of linear equations.
For the following exercises, solve the system of linear equations using Gaussian elimination.

### Solving Systems with Inverses

For the following exercises, find the inverse of the matrix.
For the following exercises, find the solutions by computing the inverse of the matrix.
For the following exercises, write a system of equations to solve each problem. Solve the system of equations.

### Solving Systems with Cramer's Rule

For the following exercises, find the determinant.
For the following exercises, use Cramer’s Rule to solve the linear systems of equations.

### Practice Test

Is the following ordered pair a solution to the system of equations?
For the following exercises, solve the systems of linear and nonlinear equations using substitution or elimination. Indicate if no solution exists.
For the following exercises, graph the following inequalities.
For the following exercises, write the partial fraction decomposition.
For the following exercises, perform the given matrix operations.
For the following exercises, use Gaussian elimination to solve the systems of equations.
For the following exercises, use the inverse of a matrix to solve the systems of equations.
For the following exercises, use Cramer’s Rule to solve the systems of equations.
For the following exercises, solve using a system of linear equations.