Which system of equations can be graphed to find the solution(s) to [tex]\( 2 - 9x = \frac{x}{2} - 1 \)[/tex] ?

A. [tex]\(\left\{\begin{array}{l} y = 2 - 9x \\ y = \frac{x}{2} - 1 \end{array}\right.\)[/tex]

B. [tex]\(\left\{\begin{array}{l} y = 2 - 9x \\ y = -\left(\frac{x}{2} - 1\right) \end{array}\right.\)[/tex]

C. [tex]\(\left\{\begin{array}{l} y = 9x \\ y = -\frac{x}{2} \end{array}\right.\)[/tex]

D. [tex]\(\left\{\begin{array}{l} y = -9x \\ y = \frac{x}{2} \end{array}\right.\)[/tex]



Answer :

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]