Solving Simultaneous Equations by Elimination

Solving Simultaneous Equations

Solving simultaneous equations means finding the values of the variables that satisfy all equations in the system at the same time. There are several methods to do this, and the best choice often depends on the specific equations.

Some common methods are: Substitution Method, Elimination Method and Graphical Method.

In this lesson, you will learn how to solve simultaneous equations using the Elimination Method.
Check out this other lesson on how to use the Substitution Method.
Check out this other lesson on how to use the Graphical Method.

The following diagram shows an example of solving simultaneous equations using the Elimination Method or Addition/Subtraction Method. Scroll down the page for more examples and solutions of solving simultaneous equations.


 

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Solving Simultaneous Equations using the Elimination Method
The Elimination Method involves manipulating the equations (multiplying by constants) so that when you add or subtract them, one of the variables is eliminated. This method is useful when variables have coefficients that are easy to make opposites or identical.

Steps:

1. Multiply one or both equations by a constant so that the coefficients of one of the variables are either the same or opposites.
2. Add or subtract the equations to eliminate one variable.
3. Solve the resulting single-variable equation.
4. Substitute the value found back into one of the original equations to find the value of the other variable.
5. Check your solution by substituting both values into the original equations.

Special Cases:

1. No Solution: If the lines are parallel and distinct, there is no solution. When solving algebraically, you’ll end up with a false statement.
   
2. Infinite Solutions: If the lines are the same (coincident), there are infinitely many solutions. When solving algebraically, you’ll end up with a true statement.

Solving Simultaneous Equations by Elimination Example 1
This video demonstrates how to solve simultaneous equations by elimination.
Example:
4x + 3y = 17
7x - y = 36

• Show Video Solutions

Solving Simultaneous Equations by Elimination Example 2
Example:
5x + 3y = 41
2x + 5y = 43

• Show Video Solutions

Solving Simultaneous Equations by Elimination Example 3
Example:
4x + 6y = 18
11x - 8y = -73
Eliminate y

• Show Step-by-step Solutions

Solving Simultaneous Equations by Elimination Example 4
Example:
4x + 6y = 18
11x - 8y = -73
Eliminate x

• Show Step-by-step Solutions

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