Given the following system of equations:
[tex]\begin{array}{l}
-4 x+8 y=16 \\
2 x+4 y=32
\end{array}[/tex]
What action was completed to create this new equivalent system of equations?
[tex]\begin{array}{l}
-2 x+4 y=8 \\
2 x+4 y=32
\end{array}[/tex]
A. Divide the second equation, [tex]$2 x+4 y=32$[/tex], by 2.
B. Divide the first equation, [tex]$-4 x+8 y=16$[/tex], by 2.
C. Multiply the second equation, [tex]$2 x+4 y=32$[/tex], by -1.
D. Multiply the first equation, [tex]$-4 x+8 y=16$[/tex], by -1.
Answer (2)
The first equation of the original system is divided by 2.
The second equation remains unchanged.
Therefore, the action completed is dividing the first equation by 2.
The answer is: Divide the first equation, − 4 x + 8 y = 16 , by 2 .
Explanation
Analyze the problem We are given an initial system of equations: − 4 x + 8 y = 16 2 x + 4 y = 32 And a new system of equations: − 2 x + 4 y = 8 2 x + 4 y = 32 We need to determine what action was performed on the original system to obtain the new system.
Compare the first equations Let's compare the first equation of the original system, − 4 x + 8 y = 16 , with the first equation of the new system, − 2 x + 4 y = 8 . We can see that the new equation is obtained by dividing the original equation by 2: 2 − 4 x + 8 y = 2 16 − 2 x + 4 y = 8 So, the first equation was divided by 2.
Compare the second equations Now, let's compare the second equation of the original system, 2 x + 4 y = 32 , with the second equation of the new system, 2 x + 4 y = 32 . We can see that the second equation remained unchanged.
Conclusion Therefore, the action completed to create the new equivalent system of equations was dividing the first equation, − 4 x + 8 y = 16 , by 2.
Examples
Understanding how to manipulate systems of equations is crucial in various fields, such as economics and engineering. For instance, in economics, you might have a system of equations representing supply and demand curves. By performing operations like dividing an equation by a constant, you can simplify the system and solve for equilibrium prices and quantities. This skill helps in making informed decisions and predictions about market behavior. Similarly, in engineering, manipulating equations is essential for designing circuits or analyzing structural stability.
The action completed to create the new equivalent system of equations was dividing the first equation, − 4 x + 8 y = 16 , by 2. The second equation remained unchanged. Therefore, the answer is option B.
;
