Lesson Solution of a linear system of two equations in two unknowns using determinant

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Solution of a linear system of two equations in two unknowns using determinant

Let us consider a system of two linear equations in two unknowns.
The most general form of such a system is as follows

a%5B11%5D%2Ax+%2B+a%5B12%5D%2Ay+=+b%5B1%5D

a%5B11%5D%2Ax+%2B+a%5B12%5D%2Ay+=+b%5B1%5D

, (1)
a%5B21%5D%2Ax+%2B+a%5B22%5D%2Ay+=+b%5B2%5D

, (2)

where a%5B11%5D

are coefficients; b%5B1%5D

are right side constants;

and

are unknowns.
Constants a%5B11%5D

and

are given, the unknown values

and

have to be found to satisfy equations (1) and (2).

There are several different methods to solve the linear system of equations (1), (2).
Substitution method is described in the lesson Solution of the linear system of two equations in two unknowns by the Substitution method in this module.
Elimination method is described in the lesson Solution of the linear system of two equations in two unknowns by the Elimination method.
In this lesson we describe the determinant method for the solution of the linear equation system (1), (2).

Calculation formulas

Coefficients of the linear equation system (1), (2) form the matrix %28matrix%282%2C2%2C+a%5B11%5D%2C+a%5B12%5D%2C+a%5B21%5D%2C+a%5B22%5D%29%29

(2x2 matrix with 2 rows and 2 columns),
while the right side coefficients b%5B1%5D

form the vector %28matrix%282%2C1%2C+b%5B1%5D%2C+b%5B2%5D%29%29

(which you can consider as the matrix with 2 rows and 1 column).
The determinant of the 2x2 matrix %28matrix%282%2C2%2C+a%5B11%5D%2C+a%5B12%5D%2C+a%5B21%5D%2C+a%5B22%5D%29%29

is the value

. (3)
If the determinant

of the matrix %28matrix%282%2C2%2C+a%5B11%5D%2C+a%5B12%5D%2C+a%5B21%5D%2C+a%5B22%5D%29%29

is not equal to zero, then the solution of the linear equation system (1), (2) does exist, is unique and
is expressed by the formulas

, (4)

. (5)

Note that the numerator of formulas (4), (5) is the determinant of the matrix obtained from the basic matrix replacing the first column by the right side vector
for

, and replacing the second column by the right side vector for

. So, you can rewrite formulas (4) and (5) in the form

, (4b)

, (5b)

Example

Solve the system of the two equations

system+%282x+%2B+y+=+++5%2C%0D%0A++++++++++-4x+%2B+6y+=+-2%0D%0A%29

First calculate the determinant of the matrix %28matrix%282%2C2%2C2%2C1%2C-4%2C6%29%29

.

Since the determinant is not equal to zero, then according to the theory, the solution does exist, is unique and is calculated by formulas

x+=+%285%2A6-%28-2%29%2A1%29%2Fdet+=+%2830%2B2%29%2F16+=+32%2F16+=+2

x+=+%285%2A6-%28-2%29%2A1%29%2Fdet+=+%2830%2B2%29%2F16+=+32%2F16+=+2

y+=+%282%2A%28-2%29-%28-4%29%2A5%29%2Fdet+=+%28-4-%28-20%29%29%2F16+=+16%2F16+=+1

.

Let us check the calculated solution. Simply substitute the found values of x=2

and

to the original system of equations. You will get

2%2A2%2B1+=+5

for the left side of the first equation, which coincides with its right side;
-4%2A2%2B6%2A1+=+-8%2B6+=+-2

for the left side of the second equation, which coincides with its right side.

The check is done, the solution is correct.

Checking validity of the general solution formula

We can check the validity of the general formulas (4) and (5).
To do this, simply substitute the formulas (4) and (5) to the equation (1) first.
For simplicity, let us do it for numerators only.

We have

.

After canceling the terms shown by red it is equal to
%28a%5B11%5D%2Aa%5B22%5D+-+a%5B12%5D%2Aa%5B21%5D%29%2Ab%5B1%5D

.

Canceling the numerator and denominator by the determinant, you will get exactly the right side b%5B1%5D

.
Similar proof is for the second equation. You can do it yourself.

How to derive the general solution formulas

If you want to derive the general solution formulas (4) and (5), simply apply the method "multiply and add" to solve equations (1) and (2).
Multiply the equation (1) by a%5B22%5D

, multiply the equation (2) by a%5B12%5D

and then add them to eliminate

. You will get

.

Express

from this equation, and you will get exactly the formula (4) for

.
Similar method works for

. Do it yourself please.

My other lessons on solving linear systems of two equations in two unknowns in this site are
    - Solution of a linear system of two equations in two unknowns by the Substitution method
    - Solution of a linear system of two equations in two unknowns by the Elimination method
    - Geometric interpretation of a linear system of two equations in two unknowns
    - Useful tricks when solving systems of 2 equations in 2 unknowns by the Substitution method
    - Solving word problems using linear systems of two equations in two unknowns

    - Word problems that lead to a simple system of two equations in two unknowns
    - Oranges and grapefruits
    - Using systems of equations to solve problems on tickets
    - Three methods for solving standard (typical) problems on tickets
    - Using systems of equations to solve problems on shares
    - Using systems of equations to solve problems on investment
    - Two mechanics work on a car
    - The Robinson family and the Sanders family each used their sprinklers last summer
    - Roses and vilolets
    - Counting calories and grams of fat in combined food
    - A theater group made appearances in two cities
    - Exchange problems solved using systems of linear equations
    - Typical word problems on systems of 2 equations in 2 unknowns
    - HOW TO algebreze and solve these problems on 2 equations in 2 unknowns
    - One unusual problem to solve using system of two equations
    - Non-standard problem with a tricky setup
    - Sometimes one equation is enough to find two unknowns in a unique way
    - Solving mentally word problems on two equations in two unknowns
    - Solving systems of non-linear equations by reducing to linear ones
    - Solving word problems for 3 unknowns by reducing to equations in 2 unknowns
    - System of equations helps to solve a problem for the Thanksgiving day
    - Using system of two equations to solve the problem for the day of April, 1

    - OVERVIEW of lessons on solving systems of two linear equations in two unknowns

Use this file/link ALGEBRA-I - YOUR ONLINE TEXTBOOK to navigate over all topics and lessons of the online textbook ALGEBRA-I.

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