Answered

Which system of equations can be graphed to find the solution(s) to $4 x^2=x^2+7$?


A$\begin{aligned}

y & =3 x^2+7 \\

y & =x^2+7

\end{aligned}$


B$\begin{array}{l}

y=4 x^2 \\

y=x^2+7

\end{array}$


C$\begin{array}{l}

y=4 x^2 \\

y=4 x^2+7

\end{array}$


D$\begin{array}{l}

y=3 x^2 \\

y=x^2+7

\end{array}$

Verified Answer :

  • To solve the equation \(4x^2 = x^2 + 7\) by graphing, set each side equal to \(y\) to define two separate functions.
  • The functions are \(y = 4x^2\) and \(y = x^2 + 7\).
  • The solution to the original equation corresponds to the x-coordinates of the intersection points of these two functions.
  • Comparing this result with the provided options, we identify that the correct system is B. \(\boxed{B}\)

Explanation

  1. Analyzing the Equation and Graphing Principle

To find the solutions to an equation of the form \(f(x) = g(x)\), we can represent each side of the equation as a separate function, \(y_1 = f(x)\) and \(y_2 = g(x)\). The solutions to the original equation are the x-coordinates of the points where the graphs of these two functions intersect. In our case, the given equation is \(4x^2 = x^2 + 7\). By identifying the two sides of the equation as separate functions, we obtain:
\[f(x) = 4x^2\]
\[g(x) = x^2 + 7\]
Thus, the system of equations that corresponds to this equality is:
\[\begin{aligned} y &= 4x^2 \ y &= x^2 + 7 \end{aligned}\]

  1. Evaluating the Options

We compare our derived system with the given options:

A\(\begin{aligned} y &= 3x^2 + 7 \ y &= x^2 + 7 \end{aligned}\)
B\(\begin{aligned} y &= 4x^2 \ y &= x^2 + 7 \end{aligned}\)
C\(\begin{aligned} y &= 4x^2 \ y &= 4x^2 + 7 \end{aligned}\)
D\(\begin{aligned} y &= 3x^2 \ y &= x^2 + 7 \end{aligned}\)

Option B matches our identified system exactly. Therefore, graphing the two functions \(y = 4x^2\) and \(y = x^2 + 7\) allows us to find the points of intersection, which correspond to the values of \(x\) that satisfy the equation \(4x^2 = x^2 + 7\).

  1. Conclusion

The correct system of equations is given by B. The intersection points of the parabolas \(y = 4x^2\) and \(y = x^2 + 7\) provide the solution set for \(4x^2 = x^2 + 7\). The final answer is B.

Examples

This method of graphing is widely used in economics and business to find the equilibrium point where supply equals demand. For instance, if the supply of a product is defined by a function \(S(x)\) and the demand is defined by \(D(x)\), the equilibrium price and quantity can be found by graphing \(y = S(x)\) and \(y = D(x)\). The point where these two curves meet represents the market equilibrium, helping businesses decide on optimal pricing strategies to clear their inventory efficiently.

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