Respuesta :
Answer:
see explanation
Step-by-step explanation:
The equation of a line in slope- intercept form is
y = mx + c ( m is the slope and c the y- intercept )
(1)
Rearrange 3x + 2y = 1 into this form by subtracting 3x from both sides
2y = - 3x + 1 ( divide all terms by 2 )
y = - [tex]\frac{3}{2}[/tex] x + [tex]\frac{1}{2}[/tex] ← in slope- intercept form
with slope m = - [tex]\frac{3}{2}[/tex]
Parallel lines have equal slopes, thus
y = - [tex]\frac{3}{2}[/tex] x + c ← is the partial equation
To find c substitute (- 6, 15) into the partial equation
15 = 9 + c ⇒ c = 15 - 9 = 6
y = - [tex]\frac{3}{2}[/tex] x + 6 ← equation of line
(2)
Rearrange x + 2y = 8 into slope- intercept form by subtracting x from both sides
2y = - x + 8 ( divide all terms by 2 )
y = - [tex]\frac{1}{2}[/tex] x + 4 ← in slope- intercept form
with slope m = - [tex]\frac{1}{2}[/tex]
Given a line with slope m then the slope of a line perpendicular to it is
[tex]m_{perpendicular}[/tex] = - [tex]\frac{1}{m}[/tex] = - [tex]\frac{1}{-\frac{1}{2} }[/tex] = 2, thus
y = 2x + c ← is the partial equation
To find c substitute (5, - 2) into the partial equation
- 2 = 10 + c ⇒ c = - 2 - 10 = - 12
y = 2x - 12 ← equation of line
Answer: (1) y = -3x/2 + 6
(2) y = 2x - 12
Step-by-step explanation:
The slope - intercept form is given as :
y = mx + c , where m is the slope and c is the y - intercept.
Two lines are said to be parallel if they have the same slope , that is if the slope of the first line is M1 and the slope of the second line is M2 , if they are parallel, M1 = M2.
(1) the equation of the given line is 3x + 2y = 1 , to find the slope of the line parallel to it , we must write the equation in slope - intercept form.
3x + 2y = 1
make y the subject of the formula
2y = -3x + 1
y = -3x/2 + 1/2
comparing with y = mx + c , the slope is -3/2 , we can now find the equation of the second line since the slope is known and it passes through the point (-6,15) using the formula:
y - [tex]y_{1}[/tex] = m( x - [tex]x_{1}[/tex])
y - 15 = [tex]\frac{-3}{2}[/tex] ( x -{-6} )
y - 15 = [tex]\frac{-3}{2}[/tex](x + 6)
2 (y - 15) = -3 (x +6)
2y - 30 = -3x - 18
2y = -3x - 18 + 30
2y = -3x +12
y = -3x/2 + 6
This gives the equation that is parallel to 3x+2y = 1
(2) Two lines are said to be perpendicular if the product of their slope is -1 , that is if M1 is the slope of the first line and M2 is the slope of the second line , then M1.M2 = -1 , that is
[tex]m_{1}[/tex] = [tex]\frac{-1}{m_{2} }[/tex]
The equation of the line given is x + 2y = 8 , we have to write it in slope - intercept form to find the slope.
x + 2y = 8
2y = -x + 8
y = -x/2 + 4
The slope is -1/2 , the slope that is perpendicular to it is therefore 2.
We can now find the equation using the formula
y - [tex]y_{1}[/tex] = m( x - [tex]x_{1}[/tex])
y - (-2) = 2 ( x - 5)
y + 2 = 2x - 10
y = 2x -10 - 2
y = 2x - 12
Therefore , the equation of line perpendicular to x + 2y = 8 is y = 2x - 12
