Question

hi ,how do i find an equation of the line that passes through the point (2, −5) and is perpendicular to the line passing through the points (−3, −2) and (5, 4). (Let x be the independent variable and y be the dependent variable.)

Answer 1

Step-by-step explanation

Since we need a line that passes through the points (2,-5) and perpendicular to the line that passes through the points (x_1,y_1) = (−3, −2) and (x_2,y_2) = (5, 4).

We first need to compute the slope of this line by applying the slope formula:

`\text{Slope}=(y_2-y_1)/(x_2-x_1)`

Substituting terms:

`\text{Slope}=(4-(-2))/(5-(-3))`

Subtracting numbers:

`\text{Slope}=(6)/(8)=(3)/(4)`

As the first line is perpendicular to the second, the slope should be negative and reciprocal.

Thus, the slope of the first line is slope= -4/3

Now, we already know that the equation of a line is as follows:

`y=mx\text{ + b}`

Where x=2 and y=-5 (the given point) and m=slope= -4/3

Plugging in the values into the equation:

`-5=-(4)/(3)\cdot2+b`

Multiplying numbers:

`-5\text{ = -}(8)/(3)+b`

Adding +8/3 to both sides:

`-(7)/(3)=b`

Finally, the equation of the line is the following:

`y=-(4)/(3)x-(7)/(3)`