Solve Linear Equation: Finding Perpendicular Line Through (0,0) to y-2x=2y

To solve this problem, we'll follow these steps:

• Simplify the given equation to find its slope.
• Determine the slope for the perpendicular line using the negative reciprocal.
• Write the equation of the line through the origin with this slope.

Let's work through each step:
Step 1: The original line is given by the equation $y - 2x = 2y$. Simplifying, we subtract $y$ from both sides to get $-2x = y$ or, equivalently, $y = -2x$. Here, the slope $m_1 = -2$.

Step 2: For the line to be perpendicular, its slope $m_2$ must satisfy $m_1 \cdot m_2 = -1$. Thus, we have $-2 \cdot m_2 = -1$, yielding $m_2 = \frac{1}{2}$.

Step 3: The equation of the line with this slope that passes through the origin is of the form $y = mx$. Substituting the slope $\frac{1}{2}$, we have $y = \frac{1}{2}x$.

Therefore, the function representing the straight line through the origin and perpendicular to the given line is $y = \frac{1}{2}x$, which corresponds to choice 1.