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The Solving Systems Of Linear Equations With The Substituting Method Calculator will calculate:

- The values of the variables x and y in any system of linear equations using the substituting method.

The x-coordinate of the solution set, x = |

The y-coordinate of the solution set, y = |

x = c_{1} b_{2} + b_{1} c_{2}/a_{1} b_{2} + a_{2} b_{1}x = × + × / × + × x = /x = |

y = ∙ (a_{2}/b_{2}) - c_{1} b_{2} + b_{1} c_{2}/a_{1} b_{2} + a_{2} b_{1}c_{2}/b_{2}y = ∙ (/) - × + × / × + × /y = ∙ () - /y = ∙ - y = |

Formula Notes: |

These formulas derive from the general form of the system of the linear equationsa_{1} x + b_{1} y = c_{1}a_{2} x + b_{2} y = c_{2} |

Solving Systems Of Linear Equations With The Substituting Method Calculator Input Values |

The coefficient preceding the variable x in the first equation (a_{1}) = |

The coefficient preceding the variable y in the first equation (b_{1}) = |

The constant in the first equation (c_{1}) = |

The coefficient preceding the variable x in the second equation (a_{2}) = |

The coefficient preceding the variable y in the second equation (b_{2}) = |

The constant in the second equation (c_{2}) = |

Please note that the formula for each calculation along with detailed calculations is shown further below this page. As you enter the specific factors of each solving systems of linear equations with the substituting method calculation, the Solving Systems Of Linear Equations With The Substituting Method Calculator will automatically calculate the results and update the formula elements with each element of the solving systems of linear equations with the substituting method calculation. You can then email or print this solving systems of linear equations with the substituting method calculation as required for later use.

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Solving a single linear equation means finding an infinity of possible pairs of values that when combined give a straight line as a graph. From geometry, it is known that a line represents a set of points, where each point represents a pair of coordinates, that if inserted in the place of the corresponding variables, make the equation true. Therefore, it is impossible to solve a single linear equation, as the number of possible solutions is infinite. The only thing we can do to find exact values for the variables in such conditions is to simultaneously solve a pair of linear equations. In this way, we (perhaps) obtain a common pair of numbers as a solution, shown graphically by the intercept point of the two lines that represent each equation involved.

In general, a system of linear equations is written in the form

For example,

is a system of linear equations, where a_{1}2121 and c_{2}.

There are three methods for solving a system of linear equations:

Elimination method, where (if necessary) the coefficients are multiplied by suitable numbers so that after adding the two equations to each other (which basically means adding the two left parts separately as well as the right parts), one of the variables is eliminated. In this way, the resulting one-variable linear equation is solved in terms of the existing variable. After finding its value, we substitute it in any of the original equations to find the other variable (the one that was eliminated before). Hence, the two values obtained for the variables (the solutions) represent a pair of numbers that when inserted into the original equations give true results.

For example, in the system of linear equations

we can multiply the first equation by -2 to get rid of the variable y after adding the two equations, i.e.

-10x + 3x - 4y + 4y = -18 + 11

-7x = -7

x = 1

Hence, substituting this value obtained for the variable x in any of the original equations (for example in the first), yields

5 ∙ 1 + 2y = 9

y = 2

y = 2

Therefore, the pair (1, 2) is a solution set for the system of linear equations.

Substitution method. This method consists of writing one of the variables in any equation of the system in term of the other variable and the expression obtained replaces the variable substituted in the other equation. In this way, a single linear equation with one variable is obtained. After solving it, the value of that variable is substituted in any of the original equations (like in the elimination method) to find the other variable. In this way, we obtain the solution set of the system of the original linear equations.

For example, in the system of linear equations

we can either express the variable x in terms of y in the first equation, i.e. writing x = 3y - 1, or express the variable y in terms of x in the second equation, i.e. y = -2x + 12. Let's use the second substitution for example. Thus, we have

x - 3(-2x + 12) = -1

x + 6x - 36 = -1

7x = 35

x = 5

x + 6x - 36 = -1

7x = 35

x = 5

Thus,

2 ∙ 5 + y = 12

y = 2

y = 2

Therefore, the number pair (5, 2) is a solution set for the given system of linear equations.

Graphing method. This method consists of plotting the graph of the two lines that represent the equations of the system and checking the coordinates of the intercept of the two graphs. These coordinates represent the solution set of the system.

The figure below represents the graphs of the two linear equations of the same system used at (b). It is evident that the x-coordinate of the lines' intercept is 5 and the corresponding y-coordinate is 2 - the same values found through the substitution method.

The calculator provided here is very useful when you don't want to solve the system but only to know the solution set. Moreover, sometimes you are not able to identify the exact coordinates of the solution set in the graph when using any graphing software or these coordinates are given as infinite decimals but you want to have them written as fractions. This calculator is perfect for this purpose, as it calculates the solutions analytically (without involving any graph) and gives the answer as fractions.

The following Math tutorials are provided within the Equations section of our Free Math Tutorials. Each Equations tutorial includes detailed Equations formula and example of how to calculate and resolve specific Equations questions and problems. At the end of each Equations tutorial you will find Equations revision questions with a hidden answer that reveal when clicked. This allows you to learn about Equations and test your knowledge of Math by answering the revision questions on Equations.

- 9.1 - Variables, Coefficients and Constants. First Order Equations with One Variable
- 9.2 - Word Problems Involving Equations
- 9.3 - Identities
- 9.4 - Iterative Methods for Solving Equations
- 9.5 - Quadratic Equations
- 9.6 - The Quadratic Formula
- 9.7 - Systems of Linear Equations. Methods for Solving Them.
- 9.8 - Relationship between Equations in Linear Systems. Systems of Equations with One Linear and One Quadratic Equation