# Linear Systems with Addition or Subtraction

## Solve systems using elimination of one variable

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Solving Linear Systems with Elimination (Addition or Subtraction)

Suppose two extended families went to an amusement park, where adult tickets and children's tickets have different prices. If the first family had 5 adults and 8 children and paid a total of $124, and the second family had 5 adults and 12 children and paid$156, how much did each type of ticket cost? Could you write a system of equations representing this situation? If you wanted to solve the system by elimination, how would you go about doing it? In this Concept, you'll learn how to use elimination to solve a system of linear equations similar to the one representing this scenario by addition or subtraction.

### Watch This

Multimedia Link For more examples of solving linear systems with elimination, see Khan Academy Solving Systems of Equations by Elimination (12:44).

Guidance

As you noticed in the previous Concept, solving a system algebraically will give you the most accurate answer and in some cases, it is easier than graphing. However, you also noticed that it took some work in several cases to rewrite one equation before you could use the Substitution Property. There is another method used to solve systems algebraically: the elimination method.

The purpose of the elimination method to solve a system is to cancel, or eliminate, a variable by either adding or subtracting the two equations. This method works well if both equations are in standard form.

#### Example A

If one apple plus one banana costs $1.25 and one apple plus two bananas costs$2.00, how much does it cost for one banana? One apple?

Solution: Begin by defining the variables of the situation. Let \begin{align*}a=\end{align*} the number of apples and \begin{align*}b=\end{align*} the number of bananas. By translating each purchase into an equation, you get the following system:

\begin{align*}\begin{cases} a+b=1.25\\ a+2b=2.00 \end{cases}\end{align*}.

You could rewrite the first equation and use the Substitution Property here, but because both equations are in standard form, you can also use the elimination method.

Notice that each equation has the value \begin{align*}1a\end{align*}. If you were to subtract these equations, what would happen?

\begin{align*}& \qquad a + b \ =1.25\\ &\underline{\;\;\;- (a+2b=2.00)\;\;\;}\\ & \qquad \quad -b =-0.75\\ & \qquad \qquad \ b =0.75\end{align*}

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