System of Equations Calculator (2024)

Select the method, input the linear equations, and click calculate button to solve linear equations using a system of linear 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 oflinear 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.

System of Equations Calculator (1)

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 tothe situation add or subtract theequation 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.
System of Equations Calculator (2024)

FAQs

How to find out how many solutions a system of equations has? ›

A system of two equations can be classified as follows: If the slopes are the same but the y-intercepts are different, the system has no solution. If the slopes are different, the system has one solution. If the slopes are the same and the y-intercepts are the same, the system has infinitely many solutions.

How to quickly solve systems of equations? ›

To Solve a System of Equations by Elimination
  1. Write both equations in standard form. ...
  2. Make the coefficients of one variable opposites. ...
  3. Add the equations resulting from Step 2 to eliminate one variable.
  4. Solve for the remaining variable.
  5. Substitute the solution from Step 4 into one of the original equations.
Mar 3, 2024

How do you check your answer to a system of equations? ›

If you are asked if a point is a solution to an equation, we replace the variables with the given values and see if the 2 sides of the equation are equal (so is a solution), or not equal (so not a solution). A solution to a system of equations means the point must work in both equations in the system.

How do you determine the number of answers possible in an equation? ›

If we can solve the equation and get something like x=b where b is a specific number, then we have one solution. If we end up with a statement that's always false, like 3=5, then there's no solution. If we end up with a statement that's always true, like 5=5, then there are infinite solutions..

How to know if a system has no solutions? ›

A system of two linear equations has no solution if the lines are parallel. Parallel lines on a coordinate plane have the same slope and different y-intercepts (see figure 3 for an example of this).

How to tell if a system of equations has one solution without graphing? ›

If the slope of the lines is different, then the system of equations has one solution. Therefore, you can determine that a system of linear equations has one solution without graphing by comparing the slope of the lines.

Is substitution or elimination easier? ›

To sum up, substitution works in all the cases you'll encounter, while elimination only works for linear cases, but elimination tends to make life easier when it works. So if it looks linear, use elimination, but if it looks non-linear (or you're really confident you can isolate one variable easily) use substitution.

What is your favorite method to use to solve systems of equations? ›

Whenever one equation is already solved for a variable, substitution will be the quickest and easiest method. Even though you're not asked to solve, these are the steps to solve the system: Substitute y + 2 y+2 y+2 for x in the second equation. Distribute the −2 and then combine like terms.

What are three methods for solving systems of equations? ›

There are three ways to solve a system of linear equations: graphing, substitution, and elimination. The solution to a system of linear equations is the ordered pair (or pairs) that satisfies all equations in the system.

What are the three possible answers to a system of equations? ›

The three possible solutions to a system of equations are one solution, infinite solutions, or no solutions. One solution means a single point satisfies the system. Infinite solutions mean an infinite number of points satisfy the system. No solution means that no points satisfy the system.

What is the answer to a system of equations called? ›

Solution is a word that we frequently use in math, but it can mean different things depending on its context. In general, however, a solution is a value or set of values that make equations true.

What if a system of equations equals zero? ›

If yes, a system of linear equations equals to zero is called an hom*ogenous system. A hom*ogeneous system always has a solution x = 0 known as its trivial solution, as it is always satisfied by x1, x2, …., xn = 0.

What does 0 0 mean in systems of equations? ›

If you are solving a system of equations and get to the point where you have: 0 = 0; this is an Identity (a situation that is always true). In a system of equations it means that the 2 equations are actually the same line (visualize one line sitting right on top of the other).

How many solutions can a system of 3 equations have? ›

An infinite number of solutions can result from several situations. The three planes could be the same, so that a solution to one equation will be the solution to the other two equations. All three equations could be different but they intersect on a line, which has infinite solutions.

How many solutions does the equation have if? ›

If the two or more equation are coincident then it has infinitely many solutions, if parallel then no solution and if intersecting then one solution.

How many solutions does this system have: 2x y = 3, 6x = 9, 3y? ›

Answer: The system of equation has infinite solution. C is correct. This is true statement.

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