Solve the system of linear equations:
$\begin{array}{l}
\frac{1}{4} x+\frac{1}{5} y=1 \\
\frac{1}{8} x-\frac{1}{8} y=1 \\
x=-12 \\
y=20
\end{array}$
Answer (1)
Substitute x = − 12 and y = 20 into the first equation: 4 1 ( − 12 ) + 5 1 ( 20 ) = 1 , which is true.
Substitute x = − 12 and y = 20 into the second equation: 8 1 ( − 12 ) − 8 1 ( 20 ) = − 4 , which is not equal to 1.
Since the second equation is not satisfied, the given values are not a solution to the system.
Therefore, the given solution is not valid: F a l se .
Explanation
Problem Analysis We are given a system of linear equations and a proposed solution x = − 12 and y = 20 . Our task is to verify whether these values satisfy the given equations.
Checking the First Equation Let's substitute x = − 12 and y = 20 into the first equation: 4 1 x + 5 1 y = 1
4 1 ( − 12 ) + 5 1 ( 20 ) = − 3 + 4 = 1 The first equation is satisfied.
Checking the Second Equation Now, let's substitute x = − 12 and y = 20 into the second equation: 8 1 x − 8 1 y = 1 8 1 ( − 12 ) − 8 1 ( 20 ) = 8 − 12 − 20 = 8 − 32 = − 4 Since − 4 = 1 , the second equation is not satisfied.
Conclusion Since the given values x = − 12 and y = 20 satisfy the first equation but not the second equation, they are not a solution to the system of equations.
Examples
Imagine you're managing a small business and need to determine the optimal pricing for your products to maximize profit. By setting up a system of equations that represents your costs and revenue, you can solve for the prices that meet your financial goals. Verifying the solution ensures that your pricing strategy aligns with your business model and helps you make informed decisions about your product offerings.
