Solve the following system of equations by using the substitution method. Check your solutions.
[tex]\left\{\begin{aligned}
x+y & =5 \\
xy & =-36
\end{aligned}\right.[/tex]
Select the correct choice below and fill in the answer boxes, if necessary.
A. The solution set is { }. (Type an ordered pair. Use a comma to separate answers as needed.)
B. There is no solution.
Answer (2)
Solve the first equation for x : x = 5 − y .
Substitute into the second equation: ( 5 − y ) y = − 36 .
Rearrange to quadratic form: y 2 − 5 y − 36 = 0 .
Solve for y : y = 9 or y = − 4 , then find corresponding x values. The solution set is {( − 4 , 9 ) , ( 9 , − 4 )} .
Explanation
Analyze the problem We are given the system of equations:
{ x + y = 5 x y = − 36
We will solve this system using the substitution method.
Solve for x First, solve the first equation for x in terms of y :
x = 5 − y
Substitute into second equation Substitute this expression for x into the second equation:
( 5 − y ) y = − 36
Rearrange to quadratic form Expand and rearrange the second equation to form a quadratic equation in y :
5 y − y 2 = − 36 ⇒ y 2 − 5 y − 36 = 0
Solve for y Solve the quadratic equation for y by factoring:
( y − 9 ) ( y + 4 ) = 0
So, y = 9 or y = − 4 .
Solve for x For each value of y , substitute it back into the equation x = 5 − y to find the corresponding value of x .
If y = 9 , then x = 5 − 9 = − 4 .
If y = − 4 , then x = 5 − ( − 4 ) = 5 + 4 = 9 .
Check the solutions Therefore, the solutions are ( − 4 , 9 ) and ( 9 , − 4 ) .
We can check our solutions by substituting them back into the original equations:
For ( − 4 , 9 ) :
x + y = − 4 + 9 = 5 x y = ( − 4 ) ( 9 ) = − 36
For ( 9 , − 4 ) :
x + y = 9 + ( − 4 ) = 5 x y = ( 9 ) ( − 4 ) = − 36
Both solutions satisfy the original equations.
Final Answer The solution set is {(-4, 9), (9, -4)}.
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 equation is y = 5 x + 100 and its revenue equation is y = 25 x , solving this system will give the number of units the company needs to sell to break even. Understanding how to solve systems of equations is crucial for making informed business decisions.
The solution set for the system of equations is {(-4, 9), (9, -4)}. Both pairs satisfy the original equations. The substitution method was successfully used to find these solutions.
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