$\left\{\begin{array}{l}-x+y=4 \\ 8 x+y=-3\end{array}\right.
Which ordered pair (x, y) is the solution to the system of equations?
Answer (1)
Test each given point in both equations.
Point (1, -3): Does not satisfy either equation.
Point (-1, 3): Satisfies the first equation but not the second.
Point (1, 3): Does not satisfy either equation.
Point (-1, -3): Does not satisfy either equation.
None of the given points are solutions to the system of equations.
Explanation
Problem Analysis We are given a system of two linear equations: − x + y = 4 8 x + y = − 3 We need to find which of the given points satisfies both equations.
Testing the Points Let's test each point to see if it satisfies both equations.
Point (1, -3):
Equation 1: − ( 1 ) + ( − 3 ) = − 4 . This does not equal 4, so the point does not satisfy the first equation.
Equation 2: 8 ( 1 ) + ( − 3 ) = 5 . This does not equal -3, so the point does not satisfy the second equation.
Point (-1, 3):
Equation 1: − ( − 1 ) + ( 3 ) = 1 + 3 = 4 . This satisfies the first equation.
Equation 2: 8 ( − 1 ) + ( 3 ) = − 8 + 3 = − 5 . This does not equal -3, so the point does not satisfy the second equation.
Point (1, 3):
Equation 1: − ( 1 ) + ( 3 ) = − 1 + 3 = 2 . This does not equal 4, so the point does not satisfy the first equation.
Equation 2: 8 ( 1 ) + ( 3 ) = 8 + 3 = 11 . This does not equal -3, so the point does not satisfy the second equation.
Point (-1, -3):
Equation 1: − ( − 1 ) + ( − 3 ) = 1 − 3 = − 2 . This does not equal 4, so the point does not satisfy the first equation.
Equation 2: 8 ( − 1 ) + ( − 3 ) = − 8 − 3 = − 11 . This does not equal -3, so the point does not satisfy the second equation.
Solving the System of Equations After testing each point, we find that none of the given points satisfy both equations. However, let's solve the system of equations to find the correct solution.
Subtract the first equation from the second equation to eliminate y :
( 8 x + y ) − ( − x + y ) = − 3 − 4 8 x + y + x − y = − 7 9 x = − 7 x = − 9 7
Substitute x = − 9 7 into the first equation to find y :
− ( − 9 7 ) + y = 4 9 7 + y = 4 y = 4 − 9 7 y = 9 36 − 9 7 y = 9 29
So the solution to the system of equations is ( − 9 7 , 9 29 ) . Since this is not one of the provided options, it means there might be a typo in the original question or the provided options. However, based on the provided options, none of them are correct.
Conclusion Since none of the given points satisfy the system of equations, there must be an error in the problem statement or the provided options. However, based on our calculations and the provided options, the closest point to being a solution is (-1, 3) as it satisfies the first equation. However, it does not satisfy the second equation. Therefore, none of the provided options are correct.
Examples
Systems of equations are used in various real-world applications, such as determining the break-even point for a business, calculating the optimal mix of ingredients in a recipe, or modeling traffic flow in a city. In this case, solving the system of equations helps us find the exact values of x and y that satisfy both equations simultaneously, which can be useful in scenarios where multiple conditions must be met.
