Solve the system of equations:
[tex]
\begin{array}{l}
y=x^2+4 x-50 \\
y=4 x+50
\end{array}
[/tex]
Write the coordinates in exact form. Simplify all fractions and radicals.
( , )
( , )
Answer (2)
Set the two equations equal to each other: x 2 + 4 x − 50 = 4 x + 50 .
Simplify the equation: x 2 − 100 = 0 .
Solve for x : x = 10 and x = − 10 .
Substitute x values into y = 4 x + 50 to find corresponding y values: ( 10 , 90 ) and ( − 10 , 10 ) .
The solutions are ( 10 , 90 ) and ( − 10 , 10 ) .
Explanation
Understanding the Problem We are given a system of two equations:
y = x 2 + 4 x − 50 y = 4 x + 50
Our goal is to find the values of x and y that satisfy both equations. These values represent the points where the two graphs intersect.
Setting the Equations Equal Since both equations are equal to y , we can set them equal to each other:
x 2 + 4 x − 50 = 4 x + 50
Now, we simplify the equation by subtracting 4 x and adding 50 to both sides:
x 2 + 4 x − 50 − 4 x − 50 = 0 x 2 − 100 = 0
Solving for x We can solve this quadratic equation by factoring or by recognizing it as a difference of squares:
x 2 − 100 = ( x − 10 ) ( x + 10 ) = 0
This gives us two possible solutions for x :
x − 10 = 0 ⇒ x = 10 x + 10 = 0 ⇒ x = − 10
Solving for y Now we substitute each value of x into one of the original equations to find the corresponding y values. We'll use the simpler equation y = 4 x + 50 :
For x = 10 :
y = 4 ( 10 ) + 50 = 40 + 50 = 90
So, one solution is ( 10 , 90 ) .
For x = − 10 :
y = 4 ( − 10 ) + 50 = − 40 + 50 = 10
So, the other solution is ( − 10 , 10 ) .
Final Answer Therefore, the solutions to the system of equations are ( 10 , 90 ) and ( − 10 , 10 ) .
Examples
Systems of equations are used in many 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 + 4 x − 50 and its revenue function is y = 4 x + 50 , solving the system of equations will tell the company the production levels ( x ) at which the cost equals the revenue, which is crucial for making informed business decisions. Understanding how to solve these systems helps in optimizing various aspects of business and economics.
The solutions to the system of equations are the points where the two graphs intersect, which are (10, 90) and (-10, 10). To find these solutions, the equations were set equal to each other, simplified, and then solved for x and y. The exact coordinates where these equations meet are therefore (10, 90) and (-10, 10).
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