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System of Equations Calculator

System of Equations Calculator

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Calculation For:

\begin{cases} \text{$a_1x + b_1y = k_1$}\\ \text{$a_2x + b_2y = k_2$}\\ \end{cases}

\begin{cases} \text{$a_1x + b_1y + c_1z = k_1$}\\ \text{$a_2x + b_2y + c_2z = k_2$}\\ \text{$a_3x + b_3y + c_3z = k_3$}\\ \end{cases}

a1

b1

k1

a2

b2

k2

a1

b1

c1

k1

a2

b2

c2

k2

a3

b3

c3

k3

Method:

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Table of Content

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The system of equations calculator is available to solve linear equations of 2 and 3 linear equations.It can be difficult for us to solve the linear equations when dealing with more than 2 linear equations.
It can be quite amazing to solve the system of equations with the algebraic method. We know there are 4 techniques to solve the system of linear equations. Here we are only solving the matrix method by the system of equations calculator.

What is the System of Linear Equations?

A System of linear is a set of linear equations 2 or more than 2 normally these equations are along with two variables. Solving systems of equations of linear
Examples:
5x+6y=3
6x+9y=12
We can solve the system of equations by the system of equations calculator.

Method of Solving the Algebraic Equation:

We can solve the algebraic equation by the following major methods:

  • The graphical method
  • The algebraic method:

The Algebraic Method:

The algebraic method of the solving the linear equation is subdivided into the four major methods:

  • The substitution method
  • The elimination method
  • The cross-multiplication method
  • The matrix method

The Substitution Method:

“In the substitution method,we calculate the value of one variable from one equation and substitute it into another equation.”
The system of equations calculator swiftly finds the answer of the linear equations.The substitution method calculator turns the task simple and elaborate for us and we can find the values of “x” and “y” quickly.

The Elimination Method:

In the elimination method we are making the coefficients of the equation equal and then subtracting them to find the answer of the variables like “x” and “y”. Solving a system of linear equations can be easily calculated if we are able to make the coefficient equal.

The Cross-Multiplication Method:

The cross-multiplication method is commonly used in solving systems of equations linear in nature. The cross-multiplication method is the most simplest method to solve linear equations. This method can be used to solve a system of linear equations of 2 or 3.

The Matrix Method :

There are three basic method to solve the system of linear equation when you are solving the linear equation by the matrix method:

Cramer’s Rule:

Cramer’s rule is an important method of solving systems of equations linear in. In the Cramer’s rules we are using the determinant of the matrices.This is the main reason the Cramer’s rule is also known as the determinant of the matrices.
Solving systems of equations by Cramer’s rule.
ax+by= k
cx+dy= l
$$ \left[ \begin{array}{cc|c}a & b & k\\c & d & l\\\end{array}\right] $$
The determinant in this case is”
$$ D = \begin{vmatrix}a & b \\ c & d\\\end{vmatrix} $$
$$D_x = \begin{vmatrix} a & b \\c & d\\\end{vmatrix} $$
$$D_y = \begin{vmatrix} a & b \\c & d\\\end{vmatrix} $$
The final values of variables “x” and “y” calculated by the system of equations calculator.
$$ x = \dfrac{D_x}{D} $$
$$ y = \dfrac{D_y}{D} $$

The Cramer’s rule is widely used to solve system of equations, as it is easy to find the final result of the variables by the Cramer’s rules. The system of equations calculator is elaborating the wholesome solution of linear equations.

The Inverse matrix method:

In the inverse matrix methodology we are multiplying with the inverse of the matrix on both sides of the equation. This is a simple system of equations with the inverse matrix. It may be possible you may find difficulty in solving systems of equations. You may be amazed to experience the working style of the system of linear equations calculator.

Consider a system of Linear equation, represented as follows:
ax+by=L
cx+dy=K

$$ \begin{bmatrix}a & b\\c & d\\\end{bmatrix} \begin{bmatrix}x\\y\\\end{bmatrix} = \begin{bmatrix} L \\ K\\\end{bmatrix} $$

$$ \begin{bmatrix} a & b\\c & d\\\end{bmatrix}^1 \begin{bmatrix}a & b\\c & d\\\end{bmatrix} \begin{bmatrix}x\\y\\\end{bmatrix} = \begin{bmatrix}a & b\\c & d\\\end{bmatrix}^1 \begin{bmatrix}L\\K \\\end{bmatrix} $$

We only have to insert the values of the coefficients and the variables to find them when using the system of equations calculator.

Gaussian-Jordan elimination:

Consider this as a method to use to solve system of linear equations.We can find the reduced echelon form by the Gaussian-Jordan elimination.
The basic steps involved in the Gaussian-Jordan elimination is as follows:

  • Change the position of the two of the rows
  • Multiply one of the row with the nonzero scalar value
  • Add and subtract the all the rows

We are able to find the reduced echelon form by the Gaussian elimination calculator.
We can represents the Gaussian-Jordan elimination as follows:
Consider the linear equation:
ax+by=L
cx+dy=K
$$ \left[ \begin{array}{cc|c}a & b & L\\c & d & K\\\end{array}\right] $$

Practical Examples:

Step1:
x+3y=5
7x+9y=11
we need to place the values of the coefficients of the variables “x” and “y”. The constant values are placed in the right side matrix.
$$ \left[ \begin{array}{cc|c}1 & 3 & 5\\7 & 9 & 11\\\end{array}\right] $$
Step2:

The determinant in this case is”
$$ D = \begin{vmatrix}1 & 3 \\7 & 9\\\end{vmatrix} = -12 $$
Step3:
We need to separate the Dx and Dy values:
D_x = \begin{vmatrix}5 & 3 \\11 & 9\\\end{vmatrix} = 12
D_y = \begin{vmatrix}1 & 5 \\7 & 11\\\end{vmatrix} = -24
Step 4:
The final values of variables “x” and “y” calculated by the system of equations solver.
$$ x = \dfrac{D_x}{D} = \dfrac{12}{-12} = -1 $$
$$ y = \dfrac{D_y}{D} = \dfrac{-24}{-12} = 2 $$
x=-1, y=2

Solving equations calculator is a simple way to solve the system of linear equations by all the 3 known matrix methods.

Working of system of equations calculator:

The system of equation solvers provides the solution of 2 or 3 linear equations in most simplest and elaborative way.
Input:

    • Insert the coefficient of variables and constant.
    • Choose the type of method to solve the equation.
    • Press the calculate button

Output:

When we are using the system of linear equations calculator.It is easy to solve the system of linear equations.

    • The final value of variables displayed
    • All the steps are represented according to the various methods

FAQs:
Why do we need a system of simultaneous equations?

When we need to find the common solution of 2 or 3 linear equations.Then we need to solve them collectively and we call them the simultaneous equations, as they have a common solution. System of equations calculator readily able to find solutions of the simultaneous equations.

Can you solve the system linear equation without graphing?

Yes, you can solve the linear equation without drawing a graph, there are different methods to solve the linear equation like substitution, elimination, and the matrix method to solve the linear equation.

How do you solve a system of equations with exponents?

You can solve a system of equations with exponents if the bases of two or more exponential equations are the same.

What are the conditions to solve the system equation by elimination ?

There are certain condition to solve system of linear equations
Write both equations in standard form
Make the coefficients of one variable opposite.
Add the equations resulting from second step 2 to eliminate one variable
Solve for the remaining variable.
Substitute the solution from fourth step 4 into one of the original equations.

What is the easiest way to solve a system of equations?

To solve a system by graphing is the most simplest method to solve the linear equation.

Conclusion:

The system of equations is necessary to solve in mathematics, when we are solving the solution of 2, or 3 linear equations. We need to find their common point to find the final solutions to our problems. The system of equations calculator provides the solution of the linear equation by the matrix method.

References:

From the source of Wikipedia:Linear equation,One variable

From the source of hmhco.com:What Is a Linear Equation? ,Describing Linear Relationships