Equations of a line to say if they are parallel or perpendicular

Question

How to show which lines are parallel and which are perpendicular. Any formulas or methods to do it?

(b) Which lines are parallel? Give a reason for your answer.

(c) Which lines are perpendicular? Give a reason for your answer.

Line $1$: $y=3x-8$

Line $2$: $y=1.5x-4$

Line $3$: $y=9x+10$

Line $4$: $y=3x+10$

Line $5$: $y=-0.5x+7$

Line $6$: $y=2x+8$

Answer 1

Lines are parallel if the slopes are the same and they are not coincident. Lines are perpendicular if the slopes multiply to form $-1$. Look through the list with this in mind.

Answer 2

If we ensure each equation is in the form $y=mx+c$ then we can easily compare the gradients of the equations. (Note: in your question they are all in this form, I'm including this comment for completeness)

For lines to be parallel, the gradients will be the same (i.e. the value of $m$ is the same in each equation)

For lines to be perpendicular, the gradients will be opposite reciprocals (in other words, multiplying the two gradients ($m$ values) together will give you $-1$).

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An example from your question:

Lines $1$ and $4$ are parallel as they both have $m=3$

Lines $5$ and $6$ are perpendicular as $-0.5\times2=-1$

Answer 3

Because all of the equations are of the form $y=mx+b$, it is sufficient to determine whether the lines are parallel by looking at the slope $m$.

Lines are parallel if they have equal slope $m$, such as $y=4x+2$ and $y=4x+8$.

Lines are perpendicular if the slopes are a negative reciprocal of each other. Namely, $m_1 = -\frac{1}{m_2}$ as in the example $y=4x+2$ and $y=-\frac{1}{4}x+7$.