Find the solution(s) of the system of equations:
[tex]
\begin{array}{l}
x^2+y^2=8 \\
y=x-4
\end{array}
[/tex]
Answer (1)
Substitute y = x − 4 into x 2 + y 2 = 8 .
Simplify the equation to x 2 − 4 x + 4 = 0 .
Factor the quadratic equation to get ( x − 2 ) 2 = 0 , so x = 2 .
Substitute x = 2 into y = x − 4 to find y = − 2 . The solution is ( 2 , − 2 ) .
Explanation
Problem Analysis We are given a system of two equations:
Equation 1: x 2 + y 2 = 8
Equation 2: y = x − 4
We need to find the solution(s) (x, y) that satisfy both equations.
Substitution Substitute the expression for y from Equation 2 into Equation 1:
x 2 + ( x − 4 ) 2 = 8
Expanding the Equation Expand and simplify the equation:
x 2 + ( x 2 − 8 x + 16 ) = 8
2 x 2 − 8 x + 16 = 8
2 x 2 − 8 x + 8 = 0
Simplifying Divide the equation by 2 to simplify:
x 2 − 4 x + 4 = 0
Factoring Factor the quadratic equation:
( x − 2 ) 2 = 0
Solving for x Solve for x :
x − 2 = 0
x = 2
Solving for y Substitute x = 2 into Equation 2 to find the corresponding value of y :
y = 2 − 4
y = − 2
Verification The solution is ( 2 , − 2 ) .
Verify the solution:
( 2 ) 2 + ( − 2 ) 2 = 4 + 4 = 8 (Correct)
− 2 = 2 − 4 (Correct)
Final Answer Therefore, the solution to the system of equations is ( 2 , − 2 ) .
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 C ( x ) = x 2 + 5 and its revenue function is R ( x ) = 4 x + 1 , solving the system of equations y = x 2 + 5 and y = 4 x + 1 will give the production level x at which the company's cost equals its revenue, which is the break-even point. In this case, solving the system helps the company make informed decisions about production and pricing.
