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Related Lessons: Adding and Subtracting Matrices Find the Inverse of 2x2 Matrix Cramer's Rule 3x3 Scalar and Matrix Multiplication

Cramer's Rule for a 2x2 System (With Two Variables)| Step by Step

 

Cramer's Rule is another method that can solve systems of linear equations using determinants. If you want to go straight to the sample problems, you can check out the five (5) worked examples right away. Otherwise, I suggest that you go over the prerequisite information first.

So, let's begin!

In terms of notations, matrix is an array of numbers enclosed by square brackets while determinant is an array of numbers enclosed by two vertical bars.

Notations
square brackets indicate matrix   vertical bars indicate determinants

 

The formula to find the determinant of a 2 x 2 matrix is very straightforward. Let's take a quick review:

The Determinant of a 2 x 2 Matrix
formula to find determinant of 2 x 2 matrix
 
Quick examples to find the Determinants of a 2 x 2 Matrix

Example 1: Find the determinant of matrix matrix A = [ [1  2][3  4]].

example 1 determinant of 2x2 matrix: determinant of matrix A = (1)(4)-(2)(3)=4-6=-2

Example 2: Find the determinant of matrix matrix B = [[5  -1][2  -3]].

example 2 determinant of 2x2 matrix: determinant of matrix B = (5)(-3)-(-1)(2)=-15-(-2)=-15+2=-13

Example 3: Find the determinant of matrix matrix C = [[-1  3] [-7  -9]].

example 3: determinant of 2x2 matrix C = (-1)(-9)-(3)(-7)=9+21=30

 

After knowing how to find the determinant of a 2 x 2 matrix, you're now ready to learn the procedures on how to use Cramer's Rule. Here we go...

Cramer's Rules for the Systems of Linear Equations with Two Variables
Given a linear system linear system in cramer's rule: showing the x-column, y-column and constant column
Assign names for each matrix
coefficient matrix marix D = [[a1  b1][a2  b2]]
X - matrix matrix Dx=[[c1  b1][c2  b2]]
Y - matrix matrix Dy=[[a1  c1][a2  c2]]
Solve variable x by formula to find x value in cramer's rule
Solve variable y by formula to find y value in cramer's rule

 

Few points to consider when looking at the formula:

1) The coefficients of variables x and y use subscripted a and b, respectively. While the constant terms use subscripted c.

2) Both denominators in solving x and y are the same. They come from the columns of x and y.

3) Looking at the numerator in solving for x, the coefficients of x-column are replaced by the constant column (in red).

4) In the same manner, to solve for y, the coefficients of y-column are replaced by the constant column (in red).

 


Direction: Solve the following systems of linear equations with two variables using Cramer's Rule.

1) example 1 of system of linear equations: 4x-3y=11 , 6x+5y=7 See Solution
2) example 2 of system of linear equations: 3x+5y=-7, x+4y=-14 See Solution
3) example 3 of system of linear equations: x-4y=-9, -x+5y=11 See Solution
4) example 4 of system of linear equations: -2x+3y=-3, 3x-4y=5 See Solution
5) example 5 of system of linear equations: 5x+y=-13, 3x-2y=0 See Solution

Example 1: Solve the system with two variables by Cramer's Rule  4x-3y=11 and 6x+5y=7 

Start by extracting the three relevant matrices: coefficient, x and y. Then solve each corresponding determinant.

  • For coefficient matrix

detailed solution to find coefficient matrix cramer's rule: determinant of matrix D =(4)(5)-(-3)(6)=20+18=38

 

 

 

 

 

  • For X - matrix

detailed solution to find x- matrix cramer's rule: determinant of matrix Dx=(11)(5)-(-3)(7)=55+21=76

 

 

 

 

 

  • For Y - matrix

detailed solution to find y- matrix cramer's rule: determinant of matrix Dy=(4)(7)-(11)(6)=28-66=-38

 

 

 

 

Once all three determinants are calculated, it's time to solve for the values of x and y using the formula above.

cramer's rule final answer example 1 where x=Dx/D = 2, and y=Dy/D=-1

I can write the final answer as (x,y) = (2,−1).


Example 2: Solve the system with two variables by Cramer's Rule  3x+5y=-7 and x+4y=-14 

Setup your coefficient, x, and y matrices from the given system of linear equations. Then calculate their determinants accordingly.

tip Remember that we always subtract the products of the diagonal entries.

 

  • For the coefficient matrix (use the coefficients of both x and y variables)

determinant of matrix D = (3)(4)-(5)(1)=12-5=7

 

 

 

  • For the X - matrix (replace the x-column by the constant column)

determinant of matrix Dx = (-7)(4)-(5)(-14)=42

 

 

 

  • For the Y - matrix (replace the y-column by the constant column)

determinant of matrix Dy = (3)(-14)-(-7)(1)=-35

 

 

 

I hope you're getting comfortable computing for the determinant of a 2-dimensional matrix. To finally solve for the required variables, I get the following results...

 

x=(Dx/D)=42/7=6, and y=Dy/D=-35/7=-5

Writing the final answer in the point notation, I got (x,y) = (6,−5).


Example 3: Solve the system with two variables by Cramer's Rule x-4y=-9 and -x+5y=11

This problem can actually be solved quite easily by Elimination Method. This is because the coefficients of variable x are the "same" but only opposite in signs ( +1 and −1 ). To solve this using the elimination method, you add their corresponding columns and the x-variable goes away - leaving you with a one-step equation in y. I am mentioning this because every technique has shortcomings and it is best to pick the most efficient. Always clarify from your teacher if it is okay to use another approach when the method is not specified on a given problem .

Anyway, since we are learning how to solve by Cramer's Rule, let's go ahead and work it out with this method.

I will construct three matrices ( coefficient, x and y) and evaluate their corresponding determinants.

  • For coefficient matrix

determinant of matrix D=(1)(5)-(-4)(-1)=1

 

 

 

  • For X - matrix ( written as uppercase D with subscript x )

determinant of matrix Dx = (-9)(5)-(-4)(11)=-1

 

 

 

  • For Y - matrix (written as uppercase D with subscript y)

determinant of matrix Dy= (1)(11)-(-9)(-1)=2

 

 

After obtaining the values of the three required determinants, I will calculate x and y as follows...

final answer to example 3: x=Dx/D=-1/1=-1, and y=Dy/D=2/1=2

The final answer in point form is just (x,y) = (−1,2).


Example 4: Solve by Cramer's Rule the system with two variables -2x+3y=-3 and 3x-4y=5

Since we have gone over a few examples already, I suggest that you try this problem on your own. Then, compare your answers to the solution below.

If you get it right the first time that means you're becoming a "pro" with regards to Cramer's Rule. If you didn't, try to figure out what went wrong and learn to not commit the same error next time. This is how you become better in math. Study many kinds of problems and more importantly, do a lot of independent practice.

check markFor coefficient matrix

determinant of matrix D = (-2)(-4)-(3)(3)=-1

 

 

 

check markFor X - matrix

determinant of matrix Dx = (-3)(-4)-(3)(5)=-3

 

 

 

check markFor Y - matrix

determinant of matrix D = (-2)(5)-(-3)(3)=-1

 

 

 

You should get the answer below...

final answer to example 4: x=Dx/Dy=-3/-1=3, and y=Dy/D=-1/-1=1

 


Example 5: Solve the system with two variables by Cramer's Rule  5x+y=-13 and 3x-2y=0

For our last example, I included a zero in the constant column. Every time you see the number zero in the constant column, I highly recommend to use Cramer's Rule to solve the system of linear equations. Why? Because the calculation of the determinants for x and y matrices drastically becomes super easy. Check it out yourself!

check markFor coefficient matrix

determinant of matrix D = (5)(-2)-(1)(3)=-12

 

 

 

check markFor X - matrix

determinant of matrix Dx=(-13)(-2)-(1)(0)=26

 

 

 

check markFor Y - matrix

determinant of matrix Dy=(5)(0)-(-13)(3)=39

 

 

 

The final answer for this problem is...

 

final answer to example 5: x=Dx/D=26/-13=-2, and y=Dy/D=39/-13=-3

 


 

Practice Problems with Answers
Worksheet 1 Worksheet 2

 

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