What is one of the solutions to the following system?
[tex]\begin{array}{l}
y-3=x \\
x^2-6 x+13=y \\
(-5,2) \\
(-2,1) \\
(2,5) \\
(8,5)
\end{array}[/tex]
Answer (2)
Test each point to see if it satisfies both equations.
Point (-5, 2) does not satisfy the first equation.
Point (-2, 1) satisfies the first equation but not the second.
Point (2, 5) satisfies both equations.
Therefore, the solution is ( 2 , 5 ) .
Explanation
Testing the points 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 which of the given points satisfies both equations. Let's test each point:
Point (-5, 2):
Equation 1: 2 − 3 = − 5 ⇒ − 1 = − 5 . This is false.
Since the first equation is not satisfied, we don't need to check the second equation.
Point (-2, 1):
Equation 1: 1 − 3 = − 2 ⇒ − 2 = − 2 . This is true.
Equation 2: ( − 2 ) 2 − 6 ( − 2 ) + 13 = 1 ⇒ 4 + 12 + 13 = 1 ⇒ 29 = 1 . This is false.
Point (2, 5):
Equation 1: 5 − 3 = 2 ⇒ 2 = 2 . This is true.
Equation 2: ( 2 ) 2 − 6 ( 2 ) + 13 = 5 ⇒ 4 − 12 + 13 = 5 ⇒ 5 = 5 . This is true.
Point (8, 5):
Equation 1: 5 − 3 = 8 ⇒ 2 = 8 . This is false.
Finding the solution The point (2, 5) satisfies both equations. Therefore, (2, 5) is 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, calculating the optimal mix of products to maximize profit, or modeling supply and demand in economics. For instance, a company might use a system of equations to find the number of units they need to sell to cover their costs and start making a profit. Understanding how to solve these systems helps in making informed decisions in business and economics.
After testing the proposed points, only Point (2, 5) satisfies both equations in the system. Therefore, the solution is ( 2 , 5 ) .
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