Question

Find the slope intercept form and the point slope the line perpendicular to 4x-7y=2 going through (-6,1)

Answer 1

Slope-intercept form:
`y=-(7)/(4)x-(19)/(2)`

Point-slope-form:
`y-1=-(7)/(4)(x+6)`

Explanation:

Hi there!

We want to find the equation of the line perpendicular to the line 4x-7y=2 that goes through (-6, 1) in slope-intercept form, as well as the point-slope form

Slope-intercept form is defined as y=mx+b, where m is the slope and b is the y intercept

Point-slope form is defined as
`y-y_1=m(x-x_1)`, where m is the slope and
`(x_1, y_1)` is a point

Meanwhile, perpendicular lines have slopes that are negative and reciprocal. When they are multiplied together, the result is -1

So let's find the slope of the line 4x-7y=2

The equation of the line is in standard form, which is ax+by=c, where a, b, and c are integer coefficients a is non-negative, and a and b aren't 0

So let's find the slope of the line 4x-7y=2

One way to do that is to convert the equation of the line from standard form to slope-intercept form

Our goal is to isolate y onto one side

Subtract 4x from both sides

-7y=-4x+2

Divide both sides by -7

y=
`(4)/(7)x-(2)/(7)`

So the slope of the line 4x-7y=2 is
`(4)/(7)`

Now, we need to find the slope of the line perpendicular to it

Use this formula:
`m_1*m_2=-1`

`m_1` in this case is
`(4)/(7)`

`(4)/(7)m_2=-1`

Multiply both sides by
`(7)/(4)`

m=
`-(7)/(4)`

Let's see the equation of the perpendicular line so far in slope-intercept form:

y=
`(-7)/(4)x`+b

We need to find b now

The equation of the line passes through (-6,1), so we can use it to solve for b.

Substitute -6 as x and 1 as y

`1=-(7)/(4)*-6+b`

Now multiply

1=
`(42)/(4)+b`

Subtract 42/4 from both sides to isolate b

-19/2=b

Substitute -19/2 as b into the equation

The equation in slope-intercept form y=
`(-7)/(4)x-(19)/(2)`

Now, here's the equation in point-slope form

Recall that the slope is
`(-7)/(4)` , our point is (-6, 1), and point-slope form is
`y-y_1=m(x-x_1)`

Let's label the value of everything to avoid any confusion

`m=-(7)/(4) \\x_1=-6\\y_1=1`

Now substitute those values into the equation

`y-1=-(7)/(4)(x--6)`

We can simplify the x--6 to x+6

`y-1=-(7)/(4)(x+6)`

Hope this helps!