Solve the system of equations:
[tex]
\begin{array}{l}
y=7 x-9 \
y=7 x+7
\end{array}
[/tex]
A. (-4,1)
B. (-4,-1)
C. (4,1)
D. No solution
Answer (2)
Set the two equations equal to each other: 7 x − 9 = 7 x + 7 .
Simplify the equation by subtracting 7 x from both sides: − 9 = 7 .
Recognize that the equation − 9 = 7 is a contradiction.
Conclude that the system of equations has No solution .
Explanation
Analyze the problem We are given the following system of equations:
y = 7 x − 9 y = 7 x + 7
We want to find the values of x and y that satisfy both equations simultaneously.
Set the equations equal Since both equations are solved for y , we can set them equal to each other:
7 x − 9 = 7 x + 7
Solve for x Now, we want to solve for x . Subtract 7 x from both sides of the equation:
7 x − 9 − 7 x = 7 x + 7 − 7 x
− 9 = 7
Conclusion The equation − 9 = 7 is a contradiction. This means there is no value of x that can satisfy both equations simultaneously. Therefore, the system of equations has no solution.
Final Answer The system of equations has no solution.
Examples
Systems of equations are used in various real-life scenarios, such as determining the break-even point for a business. For example, if a company's cost function is y = 7 x + 7 (where y is the total cost and x is the number of units produced) and the revenue function is y = 7 x − 9 (where y is the total revenue and x is the number of units sold), solving the system of equations would help determine the number of units needed to be sold for the revenue to cover the costs. In this case, since there is no solution, it means the revenue will never cover the costs, regardless of the number of units sold.
The system of equations leads to the contradiction − 9 = 7 , indicating that there are no values for x and y that can satisfy both equations. Therefore, the answer is that there is no solution. The correct option is D: No solution.
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