Consider the system:
[tex]\begin{array}{l}
2 x+5 y=40 \\
6 x-y=24
\end{array}[/tex]
You decide to use the addition method to solve this system.
Which of the following steps would you take to eliminate [tex]y[/tex]?
Answer (1)
Analyze the system of equations: 2 x + 5 y = 40 and 6 x − y = 24 .
Multiply the second equation by 5 to get 30 x − 5 y = 120 .
Add the modified second equation to the first equation to eliminate y .
Conclude that multiplying the second equation by 5 is the step to eliminate y .
Explanation
Analyze the problem We are given the following system of equations:
2 x + 5 y = 40 6 x − y = 24
Our goal is to eliminate the variable y using the addition method. This means we want to manipulate the equations so that when we add them together, the y terms cancel out.
Eliminate y To eliminate y , we need the coefficients of y in both equations to be opposites. The coefficient of y in the first equation is 5, and in the second equation is -1. We can multiply the second equation by 5 to make the coefficient of y equal to -5.
Add the equations Multiplying the second equation by 5, we get:
5 ( 6 x − y ) = 5 ( 24 ) 30 x − 5 y = 120
Now our system of equations looks like this:
2 x + 5 y = 40 30 x − 5 y = 120
Now, we can add the two equations together:
( 2 x + 5 y ) + ( 30 x − 5 y ) = 40 + 120 32 x = 160
As you can see, the y terms have been eliminated.
Conclusion Therefore, the step to eliminate y is to multiply the second equation by 5.
Examples
The addition method for solving systems of equations is useful in various real-world scenarios. For instance, consider a situation where you're trying to determine the cost of two different items, say apples and bananas, based on two separate purchases. If you know that 2 apples and 5 bananas cost $40, and 6 apples minus 1 banana costs $24, you can set up a system of equations and use the addition method to find the individual cost of each apple and banana. This method helps in scenarios where direct pricing information is not available, but combined purchase details are known.
