Given the following system of equations:
[tex]
\begin{array}{l}
2 x+6 y=12 \\
4 x+3 y=15
\end{array}
[/tex]
Which action creates an equivalent system that will eliminate one variable when they are combined?
A. Multiply the second equation by -2 to get [tex]$-8 x-6 y=-30$[/tex].
B. Multiply the first equation by 2 to get [tex]$4 x+12 y=24$[/tex].
C. Multiply the second equation by -4 to get [tex]$-16 x-12 y=-60$[/tex].
D. Multiply the first equation by -4 to get [tex]$-8 x-24 y=-48[/tex]
Answer (1)
Analyze the given system of equations and the proposed actions.
Check each action to see if it leads to the elimination of a variable when the equations are combined.
Multiplying the second equation by -2 results in an equivalent system where the y variable can be eliminated.
The correct action is to multiply the second equation by -2 to get − 8 x − 6 y = − 30 .
Explanation
Analyzing the Problem We are given a system of two linear equations with two variables, x and y :
2 x + 6 y = 12 4 x + 3 y = 15
Our goal is to determine which of the provided actions will create an equivalent system of equations such that when the equations are combined (either by addition or subtraction), one of the variables is eliminated. This is a standard technique for solving systems of equations.
Checking Each Option Let's examine each of the given options:
Multiply the second equation by -2 to get − 8 x − 6 y = − 30 . If we add this to the first equation ( 2 x + 6 y = 12 ), the y terms will cancel out:
( 2 x + 6 y ) + ( − 8 x − 6 y ) = 12 + ( − 30 ) − 6 x = − 18
This eliminates y , so this is a valid option.
Multiply the first equation by 2 to get 4 x + 12 y = 24 . If we combine this with the second equation ( 4 x + 3 y = 15 ), we won't eliminate any variables directly. We could subtract the second equation from this new equation, but that would result in 9 y = 9 , which doesn't eliminate x .
Multiply the second equation by -4 to get − 16 x − 12 y = − 60 . If we combine this with the first equation ( 2 x + 6 y = 12 ), we won't eliminate any variables directly.
Multiply the first equation by -4 to get − 8 x − 24 y = − 48 . If we combine this with the second equation ( 4 x + 3 y = 15 ), we won't eliminate any variables directly.
Determining the Correct Action From the analysis above, only the first option results in the elimination of a variable when the equations are combined. Multiplying the second equation by -2 gives − 8 x − 6 y = − 30 . Adding this to the first equation 2 x + 6 y = 12 eliminates the y variable.
Final Answer Therefore, the action that creates an equivalent system that will eliminate one variable when they are combined is to multiply the second equation by -2 to get − 8 x − 6 y = − 30 .
Examples
Systems of equations are used in many real-world applications, such as determining the break-even point for a business. For example, suppose a company has fixed costs of $10,000 and variable costs of $5 per unit. If the company sells each unit for 15 , w ec an se t u p a sys t e m o f e q u a t i o n s t o f in d t h e n u mb ero f u ni t s t ha t n ee d t o b eso l d t o b re ak e v e n . L e t x b e t h e n u mb ero f u ni t s an d y b e t h e t o t a l cos t / re v e n u e . T h ecos t e q u a t i o ni s y = 5x + 10000 an d t h ere v e n u ee q u a t i o ni s y = 15x$. Solving this system will give the break-even point.
