To determine which system of equations can be graphed to find the solution(s) to [tex]\(2-9x=\frac{x}{2}-1\)[/tex], we need to start by analyzing both sides of the given equation.
Our initial step will be to isolate the variables and constants to clearly compare them:
Starting with the equation:
[tex]\[ 2 - 9x = \frac{x}{2} - 1 \][/tex]
Rewriting it in a form where both sides represent functions of [tex]\( y \)[/tex]:
On the left-hand side:
[tex]\[ y = 2 - 9x \][/tex]
On the right-hand side:
[tex]\[ y = \frac{x}{2} - 1 \][/tex]
Thus, we convert the given single equation into a system of two equations by setting separate expressions for [tex]\( y \)[/tex]:
[tex]\[ y = 2 - 9x \][/tex]
[tex]\[ y = \frac{x}{2} - 1 \][/tex]
Now we compare these equations with the given options:
1. [tex]\(\left\{\begin{array}{l}y=2-9 x \\ y=\frac{x}{2}-1\end{array}\right.\)[/tex]
2. [tex]\(\left\{\begin{array}{l}y=2-9 x \\ y=-\left(\frac{x}{2}-1\right)\end{array}\right.\)[/tex]
3. [tex]\(\left\{\begin{array}{l}y=9 x \\ y=-\frac{x}{2}\end{array}\right.\)[/tex]
4. [tex]\(\left\{\begin{array}{l}y=-9 x \\ y=\frac{x}{2}\end{array}\right.\)[/tex]
We accurately see that the system of equations in the first option exactly matches our derived equations:
[tex]\[ y = 2 - 9x \][/tex]
[tex]\[ y = \frac{x}{2} - 1 \][/tex]
Therefore, the correct system of equations that can be graphed to find the solution(s) to [tex]\(2-9 x = \frac{x}{2} - 1\)[/tex] is:
[tex]\[
\left\{\begin{array}{l}
y = 2 - 9 x \\
y = \frac{x}{2} - 1
\end{array}\right.
\][/tex]
