A system of equations and its solution are given below.
System A
[tex]\begin{aligned}
-x-2 y & =7 \\
5 x-6 y & =-3
\end{aligned}[/tex]
Solution: [tex](-3,-2)[/tex]
Choose the correct option that explains what steps were followed to obtain the system of equations below.
System B
[tex]\begin{aligned}
-x-2 y & =7 \\
-16 y & =32
\end{aligned}[/tex]
A. To get system B, the second equation in system A was replaced by the sum of that equation and the first equation multiplied by -5. The solution to system B will not be the same as the solution to system A.
B. To get system B, the second equation in system A was replaced by the sum of that equation and the first equation multiplied by 3. The solution to system [tex]B[/tex] will be the same as the solution to system [tex]A[/tex].
C. To get system [tex]B[/tex], the second equation in system [tex]A[/tex] was replaced by the sum of that equation and the first equation multiplied by 5. The solution to system B will be the same as the solution to system A.
D. To get system B, the second equation in system A was replaced by the sum of that equation and the first equation multiplied by -6. The solution to system B will not be the same as the solution to system A.
Answer (2)
The second equation in System B is obtained by adding a multiple of the first equation in System A to the second equation in System A.
We find the value of 'k' such that the 'x' term vanishes: 5 − k = 0 , which gives k = 5 .
Substituting k = 5 into the equation, we get − 16 y = 32 , which matches the second equation in System B.
Since we obtained System B by adding a multiple of one equation to another, the solution to System B will be the same as the solution to System A. The correct option is C.
Explanation
Understanding the Problem We are given two systems of equations, System A and System B, and we need to determine how System B was obtained from System A. Specifically, we need to identify the operation performed on the equations of System A to arrive at System B and whether this operation preserves the solution of the system.
Analyzing the Systems of Equations System A is given by:
− x − 2 y = 7 5 x − 6 y = − 3
System B is given by:
− x − 2 y = 7 − 16 y = 32
Notice that the first equation in both systems is the same. The question is how the second equation in System B (-16y = 32) was obtained from System A.
Expressing the Transformation Let's assume that the second equation in System B is obtained by replacing the second equation in System A with the sum of that equation and a multiple 'k' of the first equation in System A. This means we are performing the following operation:
(Second equation in System A) + k * (First equation in System A) -> New second equation
So, we have:
( 5 x − 6 y ) + k ( − x − 2 y ) = − 3 + 7 k
Simplifying this, we get:
( 5 − k ) x + ( − 6 − 2 k ) y = − 3 + 7 k
Finding the Value of k We want to find the value of 'k' such that the 'x' term vanishes in the new second equation, leaving us with only the 'y' term. This is because the second equation in System B is -16y = 32, which has no 'x' term. Therefore, we need to find 'k' such that:
5 − k = 0
Solving for 'k', we get:
k = 5
Verifying the Transformation Now, substitute k = 5 back into the equation (5-k)x + (-6-2k)y = -3 + 7k:
( 5 − 5 ) x + ( − 6 − 2 ∗ 5 ) y = − 3 + 7 ∗ 5 0 x + ( − 6 − 10 ) y = − 3 + 35 − 16 y = 32
This matches the second equation in System B.
Checking the Solution Since we obtained System B by adding a multiple of one equation to another, the solution to System B will be the same as the solution to System A. This is because adding a multiple of one equation to another does not change the solution set of the system.
Conclusion Therefore, to get System B, the second equation in System A was replaced by the sum of that equation and the first equation multiplied by 5. The solution to System B will be the same as the solution to System A. This corresponds to option C.
Examples
Systems of equations are used in various real-world applications, such as determining the optimal mix of products to maximize profit, balancing chemical equations, and modeling supply and demand in economics. The method of replacing an equation with the sum of that equation and a multiple of another equation is a fundamental technique in solving systems of equations, ensuring that the solution remains unchanged while simplifying the system.
The correct option explaining the transformation from System A to System B is C. The second equation in System B is derived by taking the second equation from System A and adding 5 times the first equation. This operation preserves the solution, making the solutions to both systems identical.
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