Question

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

Answer 1

Answer:

Slope-intercept form: $y=-\frac{7}{4}x-\frac{19}{2}$

Point-slope-form: $y-1=-\frac{7}{4}(x+6)$

Step-by-step 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=$\frac{4}{7}x-\frac{2}{7}$

So the slope of the line 4x-7y=2 is $\frac{4}{7}$

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 $\frac{4}{7}$

$\frac{4}{7}m_2=-1$

Multiply both sides by $\frac{7}{4}$

m=$-\frac{7}{4}$

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

y=$\frac{-7}{4}x$+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

$1=-\frac{7}{4}*-6+b$

Now multiply

1=$\frac{42}{4}+b$

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=$\frac{-7}{4}x-\frac{19}{2}$

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

Recall that the slope is $\frac{-7}{4}$ , 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

$m=-\frac{7}{4} \\x_1=-6\\y_1=1$

Now substitute those values into the equation

$y-1=-\frac{7}{4}(x--6)$

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

$y-1=-\frac{7}{4}(x+6)$

Hope this helps!