Is $(4,2)$ a solution to the system $y-x=2$ and $-3x-2y=-8$?
A. Yes, because $(4,2)$ is a solution to both equations.
B. Yes, because $(4,2)$ is a solution to one equation.
C. No, because $(4,2)$ is a solution to only one equation.
D. No, because $(4,2)$ is a solution to neither equation.
Answer (1)
Substitute x = 4 and y = 2 into the first equation y − x = 2 , which results in − 2 = 2 , which is false.
Substitute x = 4 and y = 2 into the second equation − 3 x − 2 y = − 8 , which results in − 16 = − 8 , which is false.
Since the point ( 4 , 2 ) does not satisfy either equation, it is not a solution to the system.
Therefore, the answer is: No, because ( 4 , 2 ) is a solution to neither equation.
Explanation
Understanding the Problem We are given the system of equations:
y − x = 2 − 3 x − 2 y = − 8
We need to check if the point ( 4 , 2 ) is a solution to this system. A point is a solution to a system of equations if it satisfies all equations in the system.
Checking the First Equation Let's substitute x = 4 and y = 2 into the first equation:
2 − 4 = 2 − 2 = 2
This is false, so the point ( 4 , 2 ) is not a solution to the first equation.
Checking the Second Equation Now, let's substitute x = 4 and y = 2 into the second equation:
− 3 ( 4 ) − 2 ( 2 ) = − 8 − 12 − 4 = − 8 − 16 = − 8
This is also false, so the point ( 4 , 2 ) is not a solution to the second equation.
Conclusion Since the point ( 4 , 2 ) does not satisfy either equation, it is not a solution to the system of equations.
Examples
Systems of equations are used in various real-world applications, such as determining the break-even point for a business. For example, if a company's cost function is C ( x ) = 5 x + 1000 and its revenue function is R ( x ) = 15 x , solving the system of equations y = 5 x + 1000 and y = 15 x will give the number of units x that need to be sold for the company to break even. This helps businesses make informed decisions about production and pricing.
