What is the solution to this system of linear equations?
[tex]
\begin{array}{l}
x-2 y=14 \
5 x+y=32
\end{array}
[/tex]
A. (3,5)
B. (6,2)
C. (8,-1)
D. (14,-18)
Answer (2)
The solution to the system of equations is ( 11 78 , − 11 38 ) , which does not match any of the provided options. Thus, none of the options A, B, C, or D are correct. This indicates that either the solution set provided should be re-evaluated or the options given in the problem might be incorrect.
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Solve the system of equations x − 2 y = 14 and 5 x + y = 32 using the elimination method.
Multiply the second equation by 2 to get 10 x + 2 y = 64 .
Add the modified second equation to the first equation to eliminate y , resulting in 11 x = 78 , so x = 11 78 .
Substitute x = 11 78 into the first equation to find y = 11 − 38 . The solution is ( 11 78 , 11 − 38 ) .
Explanation
Problem Analysis We are given a system of two linear equations:
x − 2 y = 14 5 x + y = 32
We need to find the solution to this system and check if any of the given options is the correct solution.
Solving for x Let's solve the system of equations using the substitution or elimination method. We can multiply the second equation by 2 to eliminate the y variable:
2 ( 5 x + y ) = 2 ( 32 ) 10 x + 2 y = 64
Now we have:
x − 2 y = 14 10 x + 2 y = 64
Adding the two equations, we get:
( x − 2 y ) + ( 10 x + 2 y ) = 14 + 64 11 x = 78 x = 11 78
Solving for y Now, substitute the value of x back into one of the original equations to solve for y. Let's use the first equation:
11 78 − 2 y = 14 − 2 y = 14 − 11 78 − 2 y = 11 154 − 78 − 2 y = 11 76 y = 11 − 38
Checking the Options So, the solution to the system of equations is:
x = 11 78 , y = 11 − 38
Now, let's check if any of the given options match this solution. The given options are: (3,5), (6,2), (8,-1), (14,-18)
None of these options match the solution we found. Therefore, none of the given options are correct.
Final Answer The solution to the system of equations is:
x = 11 78 , y = 11 − 38
None of the given options are correct.
Examples
Systems of linear equations are used in various real-world applications, such as determining the optimal mix of products in manufacturing, balancing chemical equations, and modeling supply and demand in economics. For instance, a company might use a system of equations to determine the number of units of two different products they need to produce to maximize their profit, given constraints on resources like labor and materials. By solving the system, they can find the production levels that satisfy all constraints and optimize their profit.
