Solve the system:
[tex]
\begin{array}{l}
y=x^2+3 x \\
y=12 x^2+3 x
\end{array}
[/tex]
A) No solution
B) (12,1)
C) (0,0)
D) (1,12)
Answer (1)
Set the two equations equal to each other: x 2 + 3 x = 12 x 2 + 3 x .
Simplify the equation: 0 = 11 x 2 .
Solve for x : x = 0 .
Substitute x = 0 into one of the original equations to find y : y = 0 . The solution is ( 0 , 0 ) .
Explanation
Analyze the problem We are given a system of two equations:
y = x 2 + 3 x
y = 12 x 2 + 3 x
We need to find the solution(s) ( x , y ) that satisfy both equations. The answer should be one of the options: A) No solution, B) ( 12 , 1 ) , C) ( 0 , 0 ) , D) ( 1 , 12 ) .
Set equations equal To solve this system of equations, we can set the two equations equal to each other:
x 2 + 3 x = 12 x 2 + 3 x
Simplify the equation Now, let's simplify the equation by subtracting x 2 + 3 x from both sides:
0 = 12 x 2 + 3 x − ( x 2 + 3 x )
0 = 12 x 2 + 3 x − x 2 − 3 x
0 = 11 x 2
Solve for x Solve for x :
11 x 2 = 0
x 2 = 0
x = 0
Solve for y Substitute x = 0 into either of the original equations to find the corresponding y value. Let's use the first equation:
y = ( 0 ) 2 + 3 ( 0 )
y = 0 + 0
y = 0
The solution Therefore, the solution is ( 0 , 0 ) .
Compare with options Compare the solution ( 0 , 0 ) with the given options. Option C is ( 0 , 0 ) .
Final Answer The solution to the system of equations is ( 0 , 0 ) .
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 y = 5 x 2 + 2 x and its revenue function is y = 10 x 2 + 2 x , solving this system of equations will give the production level x at which the company's costs equal its revenue. In this case, solving the system 5 x 2 + 2 x = 10 x 2 + 2 x leads to x = 0 , indicating the break-even point is at zero production. This concept is crucial for businesses to understand their profitability and make informed decisions.
