– $ y \ = \ – 2 x \ – \ 7 $

– $ y \ = \ – \dfrac{ x }{ 2 } \ + \ 4 $

– $ y \ = \ \dfrac{ x }{ 2 } \ – \ 1 $

– $ y \ = \ 2 x \ + \ 9 $

This question aims to develop the understanding of **straight lines** especially the concepts of **slope, intercept**, and **perpendicular lines**.

There are **many standard forms** of writing a straight line however the most commonly used one is the **slope-intercept form**. According to the slope-intercept form, **a straight line can be written as**:

\[ y \ = \ m x \ + \ c \]

Here:

– **Dependent variable** is represented by the symbol $ y $

– **Independent variable** is represented by the symbol $ x $

– **Slope** is represented by the symbol $ m $

– **Y-intercept** is represented by the symbol $ c $

The slope of an orthogonal **line** with reference to the above line is **negative of the reciprocal** of the slope of the given equation. This can be written mathematically with the help of the **following formula**:

\[ m_{ \perp } \ = \ – \dfrac{ 1 }{ m } \]

Consequently, the **equation of this line** can be expressed with the help of the following formula:

\[ y \ = \ m_{ \perp } x \ + \ d \]

Where $ d $ can be **any real number along the y-axis**. The process of finding the **perpendicular line** is further explained in the solution given below.

## Expert Answer

**Given:**

\[ 7 x \ = \ 14 y \ – \ 8 \]

**Rearranging:**

\[ 7 x \ + \ 8 \ = \ 14 y \]

\[ \Rightarrow 14 y \ = \ 7 x \ + \ 8 \]

\[ \Rightarrow y \ = \ \dfrac{ 7 x }{ 14 } \ + \ \dfrac{ 8 }{ 14 } \]

\[ \Rightarrow y \ = \ \dfrac{ x }{ 2 } \ + \ \dfrac{ 4 }{ 7 } \]

\[ \Rightarrow y \ = \ ( \dfrac{ 1 }{ 2 } ) x \ + \ ( \dfrac{ 4 }{ 7 } ) \]

**Comparing with the standard equation** $ y \ = \ m x \ + \ c $:

\[ m \ = \ \dfrac{ 1 }{ 2 } \text{ and } c \ = \ \dfrac{ 4 }{ 7 } \]

The **slope of the perpendicular line** can be calculated using the following formula $ m_{ \perp } \ = \ – \dfrac{ 1 }{ m } $:

\[ m_{ \perp } \ = \ – \dfrac{ 1 }{ ( 1/2 ) } \]

\[ \Rightarrow m_{ \perp } \ = \ – 2 \]

Using this value in the **standard line equation** $ y \ = \ m_{ \perp } x \ + \ d $:

\[ y \ = \ – 2 x \ + \ d \]

If we **assume** $ d \ = \ -7 $:

\[ y \ = \ – 2 x \ – \ 7 \]

Which is the **correct answer out of the given options**.

## Numerical Result

\[ y \ = \ – 2 x \ – \ 7 \]

## Example

Given the equation of a **line** $ y \ = \ – 10 x \ – \ 17 $, derive the equation an **orthogonal line** with the **same y-intercept**.

The required equation is:

\[ y \ = \ – \dfrac{ 1 }{ -10 } x \ – \ 17 \]

\[ \Rightarrow y \ = \ \dfrac{ 1 }{ 10 } x \ – \ 17 \]