Finding the Perpendicular Line Equation

The problem asks us to find the equation of a line that satisfies two conditions: it must be perpendicular to the line given by the equation $x - 2y = 1$, and it must pass through the point $(1, 1)$. We can solve this by first determining the slope of the given line, then finding the slope of the perpendicular line, and finally using the point-slope form to find the equation.

Step 1: Determine the Slope of the Given Line

The equation of the given line is $x - 2y = 1$. To find its slope, we need to rewrite this equation in the slope-intercept form, which is $y = mx + c$, where '$m$' represents the slope and '$c$' represents the y-intercept.

Rearranging the equation:

• Subtract '$x$' from both sides: $-2y = -x + 1$
• Divide both sides by $-2$: $y = \frac{-x}{-2} + \frac{1}{-2}$
• Simplify: $y = \frac{1}{2}x - \frac{1}{2}$

From this slope-intercept form, we can see that the slope of the given line ($m_1$) is $\frac{1}{2}$.

Step 2: Calculate the Slope of the Perpendicular Line

Two lines are perpendicular if the product of their slopes is $-1$. Let the slope of the required perpendicular line be $m_2$. Therefore, the relationship between the slopes is:

$$ m_1 \times m_2 = -1 $$

Substitute the slope of the given line ($m_1 = \frac{1}{2}$):

$$ \frac{1}{2} \times m_2 = -1 $$

To find $m_2$, multiply both sides by $2$:

$$ m_2 = -1 \times 2 $$

$$ m_2 = -2 $$

So, the slope of the line perpendicular to the given line is $-2$.

Step 3: Use the Point-Slope Form to Find the Equation

We have the slope of the required line ($m = -2$) and a point it passes through ($(x_1, y_1) = (1, 1)$). We can use the point-slope form of a linear equation, which is $y - y_1 = m(x - x_1)$.

Substitute the values:

$$ y - 1 = -2(x - 1) $$

Now, simplify the equation:

• Distribute the $-2$ on the right side: $y - 1 = -2x + 2$
• Add $1$ to both sides to isolate '$y$': $y = -2x + 2 + 1$
• Combine the constants: $y = -2x + 3$

Step 4: Compare with the Given Options

The equation we derived is $y = -2x + 3$. Let's compare this with the given options:

• Option 1: $x + 2y = 3 \implies y = -\frac{1}{2}x + \frac{3}{2}$
• Option 2: $x + y = 2 \implies y = -x + 2$
• Option 3: $y = 2x + 3$
• Option 4: $y = -2x + 3$

Our calculated equation $y = -2x + 3$ matches Option 4.