To solve systems using elimination, follow this procedure.
- Arrange both equations in standard form, placing like variables and constants one above the other.
- Choose a variable to eliminate, and with a proper choice of multiplication, arrange so that the coefficients of that variable are opposites of one another.
- Add the equations, leaving one equation with one variable.
- Solve for the remaining variable.
- Substitute the value found in Step 4 into any equation involving both variables and solve for the other variable.
- Check the solution in both original equations.
Example 1
Solve this system of equations by using elimination.
Arrange both equations in standard form, placing like terms one above the other.
Select a variable to eliminate, say y.
The coefficients of y are 5 and –2. These both divide into 10. Arrange so that the coefficient of y is 10 in one equation and –10 in the other. To do this, multiply the top equation by 2 and the bottom equation by 5.
Add the new equations, eliminating y.
Solve for the remaining variable.
Substitute for x and solve for y.
Check the solution in the original equation.
These are both true statements. The solution is
If the elimination method produces a sentence that is always true, then the system is dependent, and either original equation is a solution. If the elimination method produces a sentence that is always false, then the system is inconsistent, and there is no solution.
The elimination method is one of the most widely used techniques for solving systems of equations.
Jenn, Founder Calcworkshop®, 15+ Years Experience (Licensed & Certified Teacher)
Why?
Because it enables us to eliminate or get rid of one of the variables, so we can solve a more simplified equation.
Some textbooks refer to the elimination method as the addition method or the method of linear combination.
This is because we are going to combine two equations with addition!
Here’s how it works.
First, we align each equation so that like variables are organized into columns.
Second, we eliminate a variable.
- If the coefficients of one variable are opposites, you add the equations to eliminate a variable, as Math Planet accurately states, and then solve.
- If the coefficients are not opposites, then we multiply one or both equations by a number to create opposite coefficients, and then add the equations to eliminate a variable and solve.
Thirdly, we substitute this value back into one of the original equations and solve for the other variable.
Simple!
Solve the System of Equations by the Elimination Method
What is really important to note is that we always add the equations – never subtract!
The reason is that addition is commutative, so no matter if we switch the order of the equations we will still arrive at an appropriate answer. Subtraction is not commutative, therefore could yield an incorrect result.
So always add!
Together we work through countless examples in detail, all while creating equivalent systems using the elimination method for solving systems of linear equations.
Elimination Method (How-To) – Video
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