# Which equation has a graph perpendicular to the graph of 7x=14y-8?

– $y \ = \ – 2 x \ – \ 7$

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

– $y \ = \ \dfrac{ x }{ 2 } \ – \ 1$

– $y \ = \ 2 x \ + \ 9$

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.

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:

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

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