Solve the system of equations.
$\begin{array}{l}
y=x^2+46 x-45 \\
y=46 x+76
\end{array}$
Write the coordinates in exact form. Simplify all fractions and radicals.
Answer (1)
Set the two expressions for y equal to each other: x 2 + 46 x − 45 = 46 x + 76 .
Simplify the equation: x 2 = 121 .
Solve for x: x = ± 11 .
Substitute each value of x into y = 46 x + 76 to find the corresponding y values. The solutions are ( 11 , 582 ) , ( − 11 , − 430 ) .
Explanation
Understanding the Problem We are given a system of two equations with two variables, x and y. Our goal is to find the values of x and y that satisfy both equations. The solutions should be in exact form, with simplified fractions and radicals.
Equating the Expressions for y Since both equations are solved for y, we can set the two expressions for y equal to each other: x 2 + 46 x − 45 = 46 x + 76
Simplifying the Equation Now, let's simplify the equation by subtracting 46 x from both sides: x 2 − 45 = 76
Isolating the x^2 Term Next, we add 45 to both sides to isolate the x 2 term: x 2 = 76 + 45
x 2 = 121
Solving for x Now, we take the square root of both sides to solve for x: x = ± 121
x = ± 11
Finding the Corresponding y Values We have two possible values for x: x = 11 and x = − 11 . We will substitute each value of x into the second equation, y = 46 x + 76 , to find the corresponding y values.
For x = 11 :
y = 46 ( 11 ) + 76 = 506 + 76 = 582
For x = − 11 :
y = 46 ( − 11 ) + 76 = − 506 + 76 = − 430
Final Answer Therefore, the solutions to the system of equations are ( 11 , 582 ) and ( − 11 , − 430 ) .
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 + 46 x − 45 and its revenue function is y = 46 x + 76 , solving the system of equations will give the production levels (x) at which the company's costs equal its revenue (y), indicating the break-even points. Understanding these points is crucial for making informed business decisions.
