Solving Systems of Equations by Addition
Greg Kelly, Hanford High School, Richland, Washington
Photo by Elizabeth Kelly, 2007
Oak Alley Plantation
Vacherie, Louisiana
Solving Systems of Equations by Addition
To use this technique, we start with two linear equations in standard form:
If the terms in the problem are not in this order, start by rearranging the equations so that the terms are in this order.
In this discussion, when we use the word coefficient, we are talking about the number beside each variable.
The coefficient of x is 5.
Let’s look at some examples of how this method works:
The coefficient of y is -3.
Example 1:
Since both sides of an equation are equal, we are really adding the same thing to both sides of the equation.
We notice that the coefficients of y are the same number but with opposite signs.
We can add the equations together, and the y terms cancel out!
Now we substitute our first answer into one of the equations to solve for the second variable.
Check in the equation that you did not use last:
Ck:
This is called the Addition Method.
It is also sometimes called the Elimination Method, because you start by eliminating one of the variables.
Example 1:
Let’s look at a few more examples:
Example 2:
This time the coefficients of the x terms are the same, but we need them to have opposite signs.
We multiply one of the equations by negative 1 to change the sign of each term in the equation before we add the equations together..
Now we substitute our first answer into one of the equations to solve for the second variable.
Check in the equation that you did not use last:
Ck:
Example 2:
Example 3:
We are looking for coefficients that match but have opposite signs.
Now we can add the equations together, and the y terms will cancel out.
We substitute our first answer into one of the original equations to solve for the second variable.
We don’t see any that match, but if we multiply the second equation by 2 then the y coefficients would match and have opposite sign.
Check in the equation that you did not use last:
Ck:
In this example, we had to change one of the equations before we could add to eliminate a variable.
Example 3:
In our last example, we will change both equations before adding.
Example 4:
In this problem, we can choose which variable to eliminate.
If necessary, change the signs in one equation and then add the equations together to eliminate the first variable.
We substitute our first answer into one of the original equations to solve for the second variable.
Since x has the smallest coefficients, we will eliminate x by multiplying to get the least common multiple of 2 and 3.
Solving Systems of Equations by Addition:
If necessary, multiply one or both equations by constants so that one of the variables does have matching coefficients with opposite signs.
2.
Add the equations to eliminate a variable, then solve for the remaining variable.
3.
Using your first answer, substitute into either equation and solve for the variable that you eliminated.
4.
Check in the equation that you did not use last.
5.
1.
Look for a variable with matching coefficients that have opposite signs.
When should we solve by addition?
The addition method is best when we cannot easily solve for one of the variables to use substitution.
The addition method works for linear equations in standard form.
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