Question

Find the equation, in slope-intercept form, of the line passing through the point (2,5) and perpendicular to the line 2x + y = 7

Answer 1

`y=\displaystyle(1)/(2)x+4`

Explanation:

Hi there!

What we need to know:

• Slope-intercept form:
  `y=mx+b` where m is the slope and b is the y-intercept
• Perpendicular lines always have slopes that are negative reciprocals (examples: 1/2 and -2, 3/4 and -4/3)

1) Determine the slope (m)

`2x + y = 7`

Reorganize the given equation into slope-intercept form; subtract 2x from both sides to isolate y:

`2x + y-2x = -2x+7\\y= -2x+7`

Now, we can easily identify the slope of the line to be -2. Because perpendicular lines always have slopes that are negative reciprocals, the slope of a perpendicular line would be
`\displaystyle(1)/(2)`. Plug this into
`y=mx+b`:

`y=\displaystyle(1)/(2)x+b`

2) Determine the y-intercept (b)

`y=\displaystyle(1)/(2)x+b`

Plug in the given point (2,5) and solve for b:

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

Subtract 1 from both sides to isolate b:

`5-1=\displaystyle(1)/(2)(2)+b-1\\4=b`

Therefore, the y-intercept of the line is 4. Plug this back into
`y=\displaystyle(1)/(2)x+b`:

`y=\displaystyle(1)/(2)x+4`

I hope this helps!