Question

Consider the line y= 3/5x-3Find the equation of the line that is parallel to this line and passes through the point (3, 4).Find the equation of the line that is perpendicular to this line and passes through the point (3, 4).

Answer 1

a) y = 3/5x + 11/5

b) y = -5/3x + 9

Step-by-step explanation:
`\begin{gathered} a)\text{ }y\text{ = }(3)/(5)x\text{ - 3} \\ \text{compare with equation of line:} \\ y\text{ = mx + b} \\ m\text{ =slope, b = y-intercept} \\ m\text{ =slope = 3/5} \\ b\text{ = -3} \end{gathered}`

For a line to be parallel to another line. the slope of the 1st line will be equalt to the slope of the 2nd line:

slope of 1st line = 3/5

So, the slope of the 2nd line = 3/5

Given point: (3, 4) = (x, y)

To get the y-intercept of the second line, we would insert the slope and the point into the equation of line

`\begin{gathered} y\text{ = mx + b} \\ 4\text{ = }(3)/(5)(3)\text{ + b} \\ 4\text{ = 9/5 + b} \\ 4\text{ - }\frac{\text{9}}{5}\text{ = b} \\ (20-9)/(5)\text{ = b} \\ b\text{ = 11/5} \end{gathered}`

The equation of line parallel to y = 3/5x - 3:

`\begin{gathered} y\text{ = mx + b} \\ y\text{ = }(3)/(5)x\text{ + }(11)/(5) \end{gathered}`
`b)\text{ line perpendicular to y = 3/5x - 3}`

For a line to be perpendicular to another line, the slope of one will be the negative reciprocal of the second line

Slope of the 1st line = 3/5

reciprocal of 3/5 = 5/3

negative reciprocal = -5/3

slope of the 2nd line (perpendicular) = -5/3

We need to get the y-intercept of the perpendicular line:

`\begin{gathered} \text{given point: (3,4) = (x, y)} \\ y\text{ = mx + b} \\ m\text{ of the perpendicular = -5/3} \\ 4\text{ = }(-5)/(3)(3)\text{ + b} \\ 4\text{ = -5 + b} \\ 4\text{ + 5 = b} \\ b\text{ = 9} \end{gathered}`

The equation of line perpendicular to y = 3/5x - 3:

`\begin{gathered} y\text{ = mx + b} \\ y\text{ = }(-5)/(3)x\text{ + 9} \end{gathered}`