Question

What is the equation of the line that passes through the point (6,8) and is perpendicular to a line with the equation y= 3/2x + 5?

Answer 1

The equation of the line that passes through the point (6,8) and is perpendicular to the line with the equation y= 3/2x + 5 is y = -2/3x + 12.

Step-by-step explanation:

The equation of a line that is perpendicular to a given line can be found using the concept that the product of their slopes must be equal to -1 (since perpendicular lines have negative reciprocal slopes). The given line has the equation y = \(\frac{3}{2}\)x + 5, which means its slope is \(\frac{3}{2}\). Therefore, the slope of the line we're looking for must be \(-\frac{2}{3}\) (the negative reciprocal of \(\frac{3}{2}\)).

To find the equation of the line that passes through the point (6,8) and has a slope of \(-\frac{2}{3}\), we use the point-slope form of a line's equation, which is y - y1 = m(x - x1), where (x1, y1) is a point on the line and m is the slope. Substituting the point (6,8) and the slope \(-\frac{2}{3}\) into this formula gives us:

y - 8 = -\frac{2}{3}(x - 6)

To get the equation in slope-intercept form (y = mx + b), we need to solve for y:

1. Multiply both sides of the equation by 3 to get rid of the fraction: 3(y - 8) = -2(x - 6)
2. Distribute on both sides: 3y - 24 = -2x + 12
3. Add 24 to both sides: 3y = -2x + 36
4. Divide both sides by 3 to solve for y: y = -\frac{2}{3}x + 12

So, the equation of the line that passes through the point (6,8) and is perpendicular to the line with equation y= 3/2x + 5 is y = -2/3x + 12.

Answer 2

`y=-(2)/(3)x+12`

Step-by-step explanation:

step 1

Find the slope of the line

we know that

If two lines are perpendicular, then their slopes are opposite reciprocal (the product of their slopes is equal to -1)

The slope of the given line is
`m=(3)/(2)`

so

The slope of the line perpendicular to the given line is
`m=-(2)/(3)`

step 2

Find the equation in point slope form

`y-y1=m(x-x1)`

we have

`m=-(2)/(3)`

`point\ (6,8)`

substitute

`y-8=-(2)/(3)(x-6)`

step 3

Convert to slope intercept form

isolate the variable y

`y-8=-(2)/(3)(x-6)\\\\y-8=-(2)/(3)x+4\\\\y=-(2)/(3)x+4+8\\\\y=-(2)/(3)x+12`