Question

"Write two equations: one parallel, one perpendicular to (5,-2) y= 1/5x-3

Write in the form y=mx±b,y=mx±b (first the parallel, then the perpendicular)"

Answer 1

For parallel:

gradient is 1/5

`y = mx + c`

consider (5, -2):

`- 2 = ( (1)/(5) * 5) + c \\ - 2 = 1 + c \\ c = - 3`

`{ \boxed{ \bf{equation : y = (1)/(5)x - 3 }}}`

For perpendicular:

gradient, m1:

`m _(1) * m_(2) = - 1 \\ m _(1) * (1)/(5) = - 1 \\ \\ m _(1) = - 5`

gradient = -5

`y = mx + c \\ - 2 = (5 * - 5) + c \\ - 2 = - 25 + c \\ c = 23`

`{ \boxed{ \bf{equation :y = - 5x + 23 }}}`

Answer 2

Parallel:
`y=\displaystyle (1)/(5)x-3`

Perpendicular:
`y=-5x+23`

Explanation:

Hi there!

What we must know:

• Slope intercept form:
  `y=mx+b` where m is the slope and b is the y-intercept (the value of y when the line crosses the y-axis)
• Parallel lines always have the same slope
• Perpendicular lines always have slopes that are negative reciprocals (ex. 2 and -1/2, -6/7 and 7/6)

Finding the Parallel Line

`y=\displaystyle (1)/(5) x-3`

Given this equation, we can identify that its slope (m) is
`\displaystyle (1)/(5)`. Because parallel lines always have the same slope, the slope of the line we're currently solving for would be
`\displaystyle (1)/(5)` as well. Plug this into
`y=mx+b`:

`y=\displaystyle (1)/(5)x+b`

Now, to find the y-intercept, plug in the given point (5,-2) and solve for b:

`-2=\displaystyle (1)/(5)(5)+b\\\\-2=1+b\\-2-1=b\\-3=b`

Therefore, the y-intercept is -3. Plug this back into
`y=\displaystyle (1)/(5)x+b`:

`y=\displaystyle (1)/(5)x+(-3)\\\\y=\displaystyle (1)/(5)x-3`

Our final equation is
`y=\displaystyle (1)/(5)x-3`.

Finding the Perpendicular Line

`y=\displaystyle (1)/(5) x-3`

Again, the slope of this line is
`\displaystyle (1)/(5)`. The slopes of perpendicular lines are negative reciprocals, so the slope of the line we're solving for would be -5. Plug this into
`y=mx+b`:

`y=-5x+b`

To find the y-intercept, plug in the point (5,-2) and solve for b:

`-2=-5(5)+b\\-2=-25+b\\-2+25=b\\23=b`

Therefore, the y-intercept is 23. Plug this back into
`y=-5x+b`:

`y=-5x+23`

Our final equation is
`y=-5x+23`.

I hope this helps!