Respuesta :
Answer:
Perpendicular: y = 4x - 20
Parallel: [tex]y = -\frac{1}{4}x + \frac{11}{2}[/tex]
Step-by-step explanation:
Pre-Solving
We are given the equation -2x-8y=4
And we want to find two equations; one that is perpendicular to this line, another that is parallel. Both of the lines also contain the point (6, 4)
The equation of the line can be written in three ways:
- Point-slope form, which is [tex]y-y_1=m(x-x_1).[/tex] m is the slope and [tex](x_1,y_1)[/tex] is a point
- Slope-intercept form, which is y=mx+b, where m is the slope and b is the y intercept
- Standard form, which is ax+by=c. a, b, and c are free integer coefficients.
The problem doesn't specify which form we should write our equation in, so technically any one of the forms should be okay. For this example however, let's write our equations in slope-intercept form, as it is the most common way to do so.
Solving
Perpendicular
Slope
Perpendicular lines have slopes that multiply to get -1.
So, first we should find the slope of -2x-8y=4.
It is currently written in the format of standard form, which is ax+by=c.
There is a shortcut into finding the slope when a line is written in standard form; the slope will be -a/b.
We can label the values of the coefficients in our line.
a = -2
b = -8
Substitute these values into the formula.
m = -a/b
m = --2/-8
m= 2/-8
m= -1/4
The slope of the line is -1/4.
However, we want the slope of the line perpendicular to it.
As already stated, the slopes of lines that are perpendicular multiply to -1.
So, to find the slope of the perpendicular line:
[tex]-\frac{1}{4} m = -1[/tex]
Multiply both sides by -4.
[tex]-4(-\frac{1}{4} m) = -4(-1)[/tex]
m = 4
The slope of our line is 4.
y-intercept
We will now find the y intercept of our line.
Here is our equation so far in slope-intercept form.
y = 4x + b
We need to find the value of b.
As the equation passes through (6, 4), we can use its values to help solve for b.
Substitute 6 as x and 4 as y.
4 = 4(6) + b
Multiply
4 = 24 + b
Subtract.
-20 = b
Substitute -20 as b.
y = 4x - 20
Parallel
Slope
Parallel lines have the same slope.
Recall that the slope of -2x -8y = 4 is -1/4.
It is also the slope of the line parallel to it.
y intercept
We can plug this value of the slope into our equation.
So far, it is:
[tex]y = -\frac{1}{4}x + b[/tex]
We need to find b now.
As the equation passes through (6, 4), we can use its values to help solve for b.
Substitute 6 as x and 4 as y.
[tex]4 = -\frac{1}{4}(6) + b[/tex]
Multiply
[tex]4 = -\frac{6}{4} + b[/tex]
Add -6/4 to both sides
[tex]4 +\frac{6}{4} = b[/tex]
[tex]\frac{11}{2} = b[/tex]
Substitute 11/2 as b in the equation
[tex]y = -\frac{1}{4}x + \frac{11}{2}[/tex]
Topic: parallel and perpendicular lines.
See more: https://brainly.com/question/20377602
