Elimination Using Multiplication

Senin, 17 Juli 2023

You have learned when and how to solve systems of equations by graphing, substitution, and elimination using addition or subtraction. The best times to use these methods are summarized in the table below.

Sometimes neither of the variables in a system of equations can be eliminated by simply adding or subtracting the equations. In this case, another method is to multiply one or both of the equations by some number so that adding or subtracting eliminates one of the variables.

 

Example

 

1.    Use elimination to solve the system of equations.

2x + 3y = 9

8x – 5y = 19

 

Alternative Solutions:

 

Multiply the first equation by –4 so that the x terms are additive inverses.

Now find the value of x by replacing y with 1 in either equation.

 

2.    Use elimination to solve the system of equations.

5x – 6y = 25

4x + 2y = 3

 

Alternative Solutions:

 

Multiply the second equation by 3 so that the y terms are additive inverses.

Now find the value of y by replacing x with 2 in either equation.

The solution of the system of equations is . Check this solution.

 

Example

Transportation Link

 

3.    Chris and Alana both take the Metro train to work. In May, Chris took the train 15 times during rush hour and 29 times during nonrush hour for $64.80. Alana took the train 30 times during rush hour and 14 times during non-rush hour for $76.80. What are the rush hour and non-rush hour fares?

 

Alternative Solutions:

Multiply the first equation by –2 to eliminate the r terms.

Now find the value of r by replacing n with 1.20 in either equation.

The solution is (2, 1.20). This means that the rush hour fare is $2.00 and the non-rush hour fare is $1.20. Do these fares seem reasonable? Check this solution by substituting (2, 120) for (r, n) in the original equations.

 

Sometimes it is necessary to multiply each equation by a different number and then add in order to eliminate one of the variables.

 

Example

 

4.    Use elimination to solve the system of equations.

3x + 4y = –25

2x – 3y = 6

 

Alternative Solutions:

 

Multiply the first equation by 2 and the second equation by –3 so that the x terms are additive inverses.

Now find the value of x by replacing y with –4 in either equation.

The solution of the system of equations is (3, 4). You can also solve this system of equations by multiplying the first equation by 3 and the second equation by 4. Why?

 

Systems of equations can be used to solve rate problems.

 

Example

Transportation Link

 

5.   A barge on the Mississippi River travels 36 miles upstream in 6 hours. The return trip takes the barge only 4 hours. Find the rate of the current.

 

Alternative Solutions:

 

 

 

Sumber

Labels: Mathematician

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