System of Equations Calculator

System of Equations Calculator

System of equations calculator is a tool that is used to solve the system of linear equations simultaneously. To solve the system of linear equations, this calculator uses the substitution method and elimination method.

What is a system of equations?

A system of equations is a collection of linear equations that are to be solved simultaneously. The values of the variables that satisfy all of the equations in the system would be evaluated by solving a system of linear equations. 

There are two well-known methods to solve the system of linear equations.

• Elimination Method
• Substitution Method

What is the Substitution Method?

This method is used to solve a system of linear equations by substituting the value of one variable. Find the value of one variable (say “x”) that is dependent on another variable (say “y”) by one equation and substitute it into the other equation. To solve the system of linear equations by substitution follow the below steps.

• Take one equation from the linear system and solve it for one variable in terms of another.
• Substitute the above variable value in the other equations and eliminate the variable's value using some arithmetic operations.
• Solve the equation formed in the first step by using any of the variable values already found in the above steps.

Example of substitution method

Solve the following system of linear equations by substitution method.

x + 3y = -4

4x - y = 1

Solution:

Step 1: write the above equation and give the name eq (i) & eq (ii).

x + 3y = -4 ------> (i)

4x - y = 1--------> (ii)

Step 2: Solve the eq (i) for “x”.

x + 3y = -4

x = -4 - 3y

Step 3: Substitute the above value in eq (ii) and Simplify for “y”.

Put x = -4 -3y in (4x - y = 1)

4(-4 -3y) – y = 1

-16 – 12y - y = 1

-16 – 13y = 1

– 13y = 1 + 16

– 13y = 17

  y = 17/(-13)

y = - 17/13

Step 4: Put the above value of “y” in step 2 and simplify.

y = - 17/13 in (x = -4 -3y)

x = - 4 – 3 (-17/13)

x = - 4 + (51/13)

Solve it by taking the LCM of the right side.

x = (-52 + 51)/13

x = -1 / 13

Hence

x = -1 / 13, y = - 17/13 is the solution of the given system of linear equations.

What is the elimination method?

In this method, we eliminate the variable making the same coefficient by some arithmetic operation (multiply or divide). Some steps to eliminating the value of the variable are given below.

• Take the variables that you want to eliminate from the equation and make the coefficients of the variable the same.
• Find the LCM of selecting a variable by choosing coefficients from all equations.
• Multiply both sides of all equations with the LCM to make the same coefficients.
• According to the situation add or subtract the equation to cancel the selected variable.
• By the above step, we get the value of one variable and use this value in any equation to find the value of the eliminated variable.

Example of elimination method

Solve the following system of linear equations by elimination method.

2x – y = 3, x + 2y = 2

Solution:

Step 1: Select the variable that want to eliminate and write the above equation by assigning its name.

2x – y = 3------> (i)

x + 2y = 2------> (ii)

Eliminate the “y” to get the solution of the system in an easy way.

Step 2: Multiply by “2” with eq (i) on both sides and we get.

2(2x – y) = 2(3)

4x – 2y = 6

Step 3: Subtract the above equation with equation (ii) and simplify to eliminate the “y”.

 4x – 2y = 6

 x + 2y = 2

5x + 0y = 8

Simplify 

5x = 8

x = 8/5 = 1.6

Step 4: Put the above value of “x” in equation (ii) and simplify to find the value of “y”.

x = 8/5

x + 2y = 2

8/5 + 2y = 2

 2y = 2 – (8/5)

Take the LCM of the left side.

2y = 2 – (8/5)

2y = (10 – 8) / 5

2y = (2) / 5

y = (2) / (5)(2)

y = 2/10

y = 0.2

Hence,

x = 1.6, y = 0.2 is the solution of a given system of equations.

FAQs

How to solve linear equations by substitution?

Here are the steps to solve a system of linear equations by substitution method.

• Identify the System of Equations: There should be at least two linear equations to form a system.
• Solve One Equation for One Variable: One equation should be rearranged, as there should be one variable at the left of the equation.
• Substitute: Then substitute the equation from the above step to the second equation.
• Simplify: After substituting, the equation from step 2 the whole equation becomes one variable and constants, you have to add or subtract them to get the result of one variable.
• Substitute to Find the Other Variable: Substitute the calculated variable to any equation to get the result of the other variable.
• Write the Solution: In the end, write calculated results in ordered pairs.

How do we solve a system of equations by elimination?

Here are the steps to solve a system of linear equations by elimination method.

• First of all, write the given linear equations in standard form like Ax + By = C.
• After that, make the coefficients opposite of any variable by multiplying or dividing with a suitable digit.
• Add or subtract the adjusted equations to eliminate one of the variables. This will result in an equation with just one variable.
• The whole equation becomes one variable and constants, you have to add or subtract them to get the result of one variable.
• Then put the calculated variable to any original equation to get the result of the other variable.

How many methods are there to solve systems of equations?

There are several methods to solve a system of equations:

• Elimination Method
• Substitution Method
• Cramer’s Rule
• Matrix Method
• Iterative Methods
• Graphing Method

How is the graphing method used to solve a system of equations?

Here are the steps to solve a system of linear equations by graphing method.

1. First of all, plot the graph of the first linear equation.
2. After that, plot the second graph on the same rectangular coordinate system.
3. Find the point of intersection. Also, determine whether the lines are parallel or the same.
4. Find the solution of a system of linear equations:

• The point where two lines intersect will be the solution to the system of equations.
• The system has one unique solution if the lines intersect at one point.
• The system has no solution if the lines are parallel and never intersect. This means the equations are inconsistent.
• If the two lines coincide (are the same line), there are infinitely many solutions. Every point on the line is a solution.