System of Equations Calculator

Introduction to the System of Equations Calculator

System of Equations Calculator with steps is an online tool that helps you find the linear equation. It determines the value of variables (x,y,z) from the given system of algebraic equations in a fraction of a second.

What is a System of Equations?

A system of equations is a finite set of variables (x,y,z) in an algebraic linear equation. These types of equations can be solved with different methods to get the values of variables. It is commonly used in physics, engineering, space, etc.

Formula Behind System of Equations Solver

The three System of Equations Calculator uses the general form of system of equations to solve such problems. The System of linear equations in general form is,

\begin{matrix} a_{11} x_1 & + & a_{12} x_2 & + & … & + & a_{1n}x_n & = & b_1 \\ a_{21}x_1 & + & a_{22}x_2 & + & … & + & a_{2n}x_n & = & b_2 \\ a_{m1} & + & a_{m2}x_2 & + & … & + & a_{mn}x_n & = & b_m \end{matrix}

Whereas,

x is the variable of the nth number

ai is the coefficient of variable x

bi is the constant of n number

How to Solve Using Systems of Equations Calculator

System of Equations Solver has advanced built-in features in its server that enable you to get the value of different kinds of variable values from the given system of linear equations.

Our system of equation solver uses serval types of algebraic methods to solve various types of equation solutions. These methods are elimination, graphical representation, Gaussian elimination method, substitution method, Cramer’s rule, etc given as:

• Elimination of Variables

When you add the input value as a system of equations in the systems of equations solver it checks whether the given equation is in a standard form or not. After checking if it is not in standard form then it converts into standard form first as ax+by=c.

Let's explain the given example x+y=10, x-y=12. As you see the given equation is already in standard form, then the system of linear equations calculator applies the multiply method to equal one equation of the term from x,y to another equation.

Now it adds or subtracts the above equations to get the value of x which is x=11. After that system of equation calculator substitutes the x value in one of the given equations to get the value of y as y=-1.

Now you have both x and y values to confirm your answer is correct add both x and y values in both the equation one by one. If both the equations satisfy each other then your x and y values are right using the elimination method.

Example: Solving a System of Linear Equations using the Elimination Method

\begin{matrix} x & + & y & = & 10 \\ x & - & y & = & 12 \\ \end{matrix}

Solution:

\begin{matrix} x & + & y & = & 10 \\ x & - & y & = & 12 \end{matrix}

\begin{matrix} x & + & y & = & 10 \\ x & - & y & = & 12 \\ \hline 2x & = & 22 \end{matrix}

\begin{matrix} x & = & 11 \\ x & + & y & = & 10 \\ \hline 11 & + & y & = & 10 \end{matrix}

$$ y \;=\; -1 $$

The pair is (11,-1)

$$ x + y \;=\; 10 $$

$$ 11 + (-y) \;=\; 10 $$

$$ 10 \;=\; 10 $$

On the other hand,

$$ x - y \;=\; 12 $$

$$ 11 - (-1) \;=\; 12 $$

$$ 12 \;=\; 12 $$

• Graphical Representation

In graphical representation, we have one equation x-3y=-10. The systems of linear equations calculator First, isolate the y value, subtract the x value on both sides after that divide 3 on both sides.

After separating the y one side of the equation we have y= 1/3x+10/3. Now system of equations calculator with steps adds the supposed x value in the previous equation one by one to get the value of y.

We have three values of x and y as (x,y)=(1,11/3),(-3.7/3),(3,13/3) now represent all these points on the graph as you see in the below example.

Example: Find the system of linear equations using a graphical representation

$$ x - 3y \;=\; -10 $$

Solution:

$$ x - 3y \;=\; -10 $$

$$ -3y \;=\; -x - 10 $$

$$ y \;=\; \frac{1}{3}x + \frac{10}{3} $$

If x = 1 then,

$$ y \;=\; \frac{1}{3}(1) + \frac{10}{3} \;=\; \frac{1}{3} + \frac{10}{3} \;=\; \frac{11}{3} $$

If x = -3 then,

$$ y \;=\; \frac{1}{3}(-3) + \frac{10}{3} \;=\; -1 + \frac{10}{3} \;=\; \frac{7}{3} $$

If x = 3 then,

$$ y \;=\; \frac{1}{3}(3) + \frac{10}{3} \;=\; 1 + \frac{10}{3} \;=\; \frac{13}{3} $$

Therefore, the ordered points are (1, 11/3), (-3, 7/3), (3,13/3)

• Gaussian Elimination Method

Gaussian elimination method writes the linear equation 2x+3y=6, x-y=1/2 into an augments matrix. The gaussian elimination method used by the system of linear equations solver uses the row operation to find the value of x and y.

The solving systems of equations calculator adds and subtracts the row operation to make the matrix into echelon form. Then put the remaining term equal to zero as shown in the given example x=3/2 and y=1.

Example: Solving a System of Linear Equations using the Gaussian elimination method

\begin{matrix} 2x & + & 3y & = & 6 \\ x & - & y & = & \frac{1}{2} \end{matrix}

Solution:

\begin{array} {rrrr|ll} 2 & 3 & 6 \\ 1 & -1 & 12 \end{array}

$$ R_1 \leftrightarrow R_2 \rightarrow \begin{array} {rrrr|ll} 1 & -1 & 12 \\ 2 & 3 & 6 \end{array} $$

$$ -2R_1 + R_2 \;=\; R_2 \rightarrow \begin{array} {rrrr|ll} 1 & -1 & 1 \\ 0 & 5 & 5 \end{array} $$

$$ \frac{1}{5}R_2 \;=\; R_2 \rightarrow \begin{array} {rrrr|ll} 1 & -1 & 12 \\ 0 & 1 & 1 \end{array} $$

$$ x - (1) \;=\; frac{1}{2} $$

$$ x \;=\; \frac{3}{2} $$

So the solution is,

$$ \biggr( \frac{3}{2}, 1 \biggr) $$

• Substitution Method

In the substitution method, the system of three equations calculator first chooses the equation that is used for the substitution. Let's take two equations y=3x+6, -2x+4y=4 as you can see y is isolated so we choose the y value and substitute it in the second equation.

After putting the y value, system of equations calculator solves the equation as per the algebraic rules and gives the x value as x=-2. Then put the x value in one of the equations and we get the y value as y=0. Put the x and y values in both equations to check solution is true or not.

Example: Solving a System of Linear Equations using the Substitution method

\begin{matrix} y & = & 3x & + & 6 \\ -2x & + & 4y & = & 4 \end{matrix}

Solution:

\begin{matrix} y & = & 3x & + & 6 \\ -2x & + & 4y & = & 4 \end{matrix}

\begin{matrix} -2x & + & 4y & = & 4 \\ -2x & + & 4(3x+6) & = & 4 \end{matrix}

\begin{matrix} -2x & + & 12x & + & 24 & = & 4 \\ &&10x & + & 24 \\ &&&&-24 && 24 \\ \hline \\ &&10x &&& = & -20 \\ && x &&& = & -2 \end{matrix}

$$ y \;=\; 3x + 6 $$

$$ y \;=\; 3(-2) + 6 $$

$$ y \;=\; -6 + 6 $$

$$ y \;=\; 0 $$

$$ y \;=\; 3x + 6 \; \; \; \; \; \; \; -2x + 4y \;=\; 4 $$

$$ 0 \;=\; -6 + 6 \; \; \; \; \; \; 4 + 0 \;=\; 4 $$

$$ 0 \;=\; 0 \; \; \; \; \; 4 \;=\; 4 $$

So,

$$ x \;=\; 2 \; and\; y \;=\; 0 $$

• Cramer's Rule

In crammer rule, system of equations solver a¹x+b¹y=c¹,a²x+b²y=c² write into matric form as per the given rule x=Dx/D, y=Dy/D as shown in the given example. After solving it we have the system of equation solver gives x and y values.

Example: Solving a System of Linear Equations using the Cramer's rule

\begin{matrix} 12x & + & 3y & = & 15 \\ 2x & - & 3y & = & 13 \end{matrix}

Solution:

$$ x \;=\; \frac{D_x}{D} $$

$$ \frac{\biggr[ \begin{matrix} 15 & 3 \\ 13 & -3 \end{matrix} \biggr]}{\biggr[ \begin{matrix} 12 & 3 \\ 2 & -3 \end{matrix} \biggr]} $$

$$ \frac{-45 - 39}{-36 - 6} $$

$$ \frac{-84}{-42} $$

$$ 2 $$

Now solve for y,

$$ y \;=\; \frac{D_y}{D} $$

$$ \frac{\biggr[ \begin{matrix} 12 & 15 \\ 2 & 13 \end{matrix} \biggr]}{\biggr[ \begin{matrix} 12 & 3 \\ 2 & -3 \end{matrix} \biggr]} $$

$$ \frac{156 - 30}{-36 - 6} $$

$$ -\frac{126}{42} $$

$$ \;=\; -3 $$

Stepwise Guide for Using System of Equations Calculator

The systems of equations calculator has a simple design tool that enables you to use it to calculate the two algebraic equations with different methods.

Before entering the input function into the systems of equations solver, you must follow some simple steps so that you get a smooth experience during the calculation. These steps are:

1. Enter the first linear equation of two variables in its respective input box.
2. Enter the second linear equation of two variables in the system of linear equations calculator’s input box.
3. Review your algebraic equation of two variables before hitting the calculate button to start the evaluation process.
4. Click the “Calculate” button to get the result of your given algebraic equation problem.
5. If you want to try out our system of equation calculator for the first time then you can use the load example to get a better understanding
6. Click on the “Recalculate” button to get a refresh page for more solutions of linear equation problems

Final Output from System of Equation Solver

A three system of equations calculator gives you the solution to a given linear equation problem when you enter the input function to it. It provides you with solutions with a detailed procedure instantly. It may contain as:

• Result Option

Result option of systems of linear equations calculator gives you a solution for the linear equation to find the given algebraic equation problem

• Possible Steps

It provides you with a solution where all the evaluation processes are in a step-by-step way of the system of equation problem when you click on this option.

Advantages of Systems of Equations Solver

The system of equations solver provides you with multiple benefits whenever you use it to calculate the system of equation problems and gives you a solution. These benefits are:

• Systems of equations calculator is a free-of-cost tool, you can use it anytime to find the linear equation problem in real time.
• It is a versatile tool that allows you to get the solution of various types of algebraic equation problems
• You can try out our System of equations calculator to practice more examples of the equation of the system so that you get a strong hold on this concept
• Our tool saves you time and effort from doing linear equation calculations.
• System of linear equations calculator is a trustworthy tool that provides you with accurate solutions whenever you use it to calculate algebraic equation examples without any manmade error.
• System of equation calculator provides the solution with a complete process in a stepwise manner so that you get clarity on the linear equation problems