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Which system of equations can be graphed to find the solution(s) to [tex]\( 4x^2 = x^2 + 7 \)[/tex]?

A.
[tex]\[
\begin{cases}
y = 3x^2 \\
y = x^2 + 7
\end{cases}
\][/tex]

B.
[tex]\[
\begin{cases}
y = 3x^2 + 7 \\
y = x^2 + 7
\end{cases}
\][/tex]

C.
[tex]\[
\begin{cases}
y = 4x^2 \\
y = x^2 + 7
\end{cases}
\][/tex]

D.
[tex]\[
\begin{cases}
y = 4x^2 \\
y = 4x^2 + 7
\end{cases}
\][/tex]


Sagot :

To identify which system of equations can be graphed to find the solution(s) to [tex]\(4x^2 = x^2 + 7\)[/tex], we need to follow these steps:

1. Rearrange the given equation:
[tex]\[ 4x^2 = x^2 + 7 \][/tex]
By rearranging, we subtract [tex]\(x^2\)[/tex] from both sides:
[tex]\[ 4x^2 - x^2 = 7 \][/tex]
This simplifies to:
[tex]\[ 3x^2 = 7 \][/tex]

2. Express the given equation as two separate equations:
We can rewrite the rearranged equation [tex]\(3x^2 = 7\)[/tex] using function notation. Essentially, we want two functions [tex]\(y\)[/tex] which form the equality:
[tex]\[ y = 4x^2 \quad \text{and} \quad y = x^2 + 7 \][/tex]

3. Select the correct system of equations:
Looking at the systems provided, we compare:

- [tex]\(\left\{\begin{array}{l}y = 3x^2 \\ y = x^2 + 7\end{array}\right.\)[/tex]
- [tex]\(\left\{\begin{array}{l}y = 3x^2 + 7 \\ y = x^2 + 7\end{array}\right.\)[/tex]
- [tex]\(\left\{\begin{array}{l}y = 4x^2 \\ y = x^2 + 7\end{array}\right.\)[/tex]
- [tex]\(\left\{\begin{array}{l}y = 4x^2 \\ y = 4x^2 + 7\end{array}\right.\)[/tex]

The system that matches [tex]\(y = 4x^2\)[/tex] and [tex]\(y = x^2 + 7\)[/tex] is:
[tex]\[ \left\{\begin{array}{l} y = 4x^2 \\ y = x^2 + 7 \end{array}\right. \][/tex]

Thus, the system of equations that can be graphed to find the solution to [tex]\(4x^2 = x^2 + 7\)[/tex] is:

[tex]\(\boxed{\left\{\begin{array}{l}y=4 x^2 \\ y=x^2+7\end{array}\right.}\)[/tex]