What is one of the solutions to the following system?
[tex]\begin{array}{l}
y-3=x \
x^2-6 x+13=y\end{array}[/tex]
A. (8,5)
B. (-5,2)
C. (2,5)
D. (-2,1)
Answer (2)
We have a system of equations and solve for y in the first equation: y = x + 3 .
Substitute y into the second equation: x 2 − 6 x + 13 = x + 3 .
Simplify and factor the quadratic equation: x 2 − 7 x + 10 = ( x − 2 ) ( x − 5 ) = 0 , which gives x = 2 or x = 5 .
Find the corresponding y values: If x = 2 , y = 5 ; if x = 5 , y = 8 . Thus, one solution is ( 2 , 5 ) .
Explanation
Understanding the Problem We are given a system of two equations:
Equation 1: y − 3 = x Equation 2: x 2 − 6 x + 13 = y
We need to find one of the solutions from the given options: (8, 5), (-5, 2), (2, 5), (-2, 1).
Substitution Let's solve the system of equations by substitution. From Equation 1, we have y = x + 3 . Substitute this into Equation 2:
x 2 − 6 x + 13 = x + 3
Simplifying to Quadratic Form Rearrange the equation to form a quadratic equation:
x 2 − 6 x − x + 13 − 3 = 0 x 2 − 7 x + 10 = 0
Solving for x Factor the quadratic equation:
( x − 2 ) ( x − 5 ) = 0
So, the solutions for x are x = 2 and x = 5 .
Solving for y Now, find the corresponding y values using y = x + 3 :
If x = 2 , then y = 2 + 3 = 5 . So, one solution is ( 2 , 5 ) .
If x = 5 , then y = 5 + 3 = 8 . So, another solution is ( 5 , 8 ) .
Finding the Matching Option Check the given options to see which one matches our solutions. We found the solutions ( 2 , 5 ) and ( 5 , 8 ) . The option ( 2 , 5 ) is among the given choices.
Final Answer Therefore, one of the solutions to the system of equations is ( 2 , 5 ) .
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 y = x 2 − 6 x + 13 (where y is the total cost and x is the number of units produced) and the revenue function is y = x + 3 , solving the system of equations will give the production levels where the cost equals the revenue, indicating the break-even points. This helps in making informed business decisions.
One of the solutions to the system of equations is (2, 5), which corresponds to option C. This was found by substituting and solving the equations to determine the values of x and y. The process involved rearranging and factoring the quadratic equation derived from the given equations.
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