equation of the line

Question

1)write an equation of the line parallel  to x+2y=6 through (8,3)
2)find the equation of a line containing points (-2,8) and (3,5)
3)write an equation of the line parallel  to y=3x-2 through (1,-3)
4)write an equation of the line perpendicular  to x+y=-3 through (0,4)
5)write an equation of the line parallel  to 2x-y=1 through (-1,2)
6)write an equation of the line parallel  to x-y=-4 through (1,-3)

Answer 1

Slope intercept form line equation is y = mx + b. where m = slope and b is y -intercept.

 Step 1. First find the slope of the line.

 The line is parallel to the given line x+2y = 6

Rewrite the equation in slope-intercept form to find the slope m

x+2y = 6

Apply subtraction property ,subbtract x from each sides.

x+2y-x = 6-x

2y = 6-x

apply division property ,divide by 2 to each sides

2y/2 =(6-x)/2

y =(6/2)-(x/2)

y = 3 +(-1/2)x

y = (-1/2)x+3

slope of the line =-1/2

parlell line has same slope, so the slope of the parlall line = -1/2

step 2 To find y intercept  

 y = mx +b                       here    (x,y) =(8,3) ; m = -1/2

3 = (-1/2)8+b

3 =-4+b

Apply addition property ,add 4 to each sides.

3+4 = -4+4+b

7 = b

⇒b = 7

step 3

Final line eqation is y =mx+b        here m= -1/2,b=7

Equation of the line parallel  to x+2y=6 through (8,3) is y  = (-1/2)x + 7

 

Answer 2

2)find the equation of a line containing points (-2,8) and (3,5)

Put the given points (x1,y1) = (-2,-8)

 (x2,y2) = (3,5)

Equation of a line containing points  is x-x1 =(y2-y1/x2-x1)(y-y1)

x-(-2) =[(5-(-8))/(3-(-2)](y-(-8))  simplification

x+2 =(5+8/3+2)(y+8)

x+2 = 13/5(y+8)

Apply multiplication property, multiply by 5 each sides

5x+10 =13(y+8)

5x+10 = 13y+104

Apply subtraction property, subtract 10 from each sides.

5x+10-10 = 13y+104-10

5x =13y+94

5x-13y-94 = 0

 

Comments

Slope-intercept form line equation is y = mx + b, where m is slope and b is y-intercept.

Let the points are (x₁, y₁) = (- 2, 8) and (x₂, y₂) = (3, 5).

Slope (m) = [(y₂ - y₁)/(x₂ -x₁)]

m = [(5 - 8)/(3 - (- 2))]

m  = [- 3/(3 + 2)]

⇒ m = - 3/5.

Now, the line equation is y = (- 3/5)x + b.

Find the y - intercept by substituting any point in the line equation say (x, y) = (3, 5).

5 = (- 3/5)(3) + b

b = 5 + (9/5)

b = (25 + 9)/5

⇒ b = 34/5.

The line equation is y = (- 3/5)x + (34/5).

Answer 3

3)write an equation of the line parallel  to y=3x-2 through (1,-3)
Slope intercept form line equation is y = mx+b

y =3x -2

Slope m =3

Parlell line has same slope, so the slope of the parlell line is 3

Next find the y intercept,

y = mx+b       here (x,y) = (1,-3) m=3

1 =3(-3)+b

1 =-6+b

1+6 = b

⇒b =7

final line eqation y = mx+b          here m=3,b=7

Equation of parlell line to y = 3x-2 through (0,-4) is

y =3x+7

Comments

Find the y - intercept by substituting the point in the parallel line equation say (x, y) = (1, - 3).

- 3  = (3)(1) + b

b = - 3 - 3

⇒ b = - 6.

The parallel line equation is y = 3x - 6.

Answer 4

4).

The line equation is x + y = - 3.

Slope - intercept form line equation is y = mx + b, where m is slope and b is y - intercept.

Write the equation in slope - intercept form.

y = - x - 3.

Compare the equation with slope - intercept form.

Slope (m) = - 1.

Because the slopes of perpendicular lines are negative reciprocals, the slope of perpendicular is 1.

Now, the perpendecular line equation is y = x + b.

Find the y - intercept by substituting the given point in the perpendecular line equation say (x, y) = (0, 4).4

4 = (1)(0) + b

⇒ b = 4.

The perpendecular line equation  is y = x + 4.

Answer 5

5).

The line equation is 2x - y = 1.

Write the equation in slope - intercept form.

y = 2x - 1

Comapare the equation with slope - intercept form.

Slope (m) = 2

Because the parallel lines have same slopes, the slope of parallel line through the point (- 1, 2) is 2.

Now the parallel line equation is y = 2x + b.

Find the y - intercept by substituting the point in the parallel line equation say (x, y) = (- 1, 2).

2  = (2)(- 1) + b

b = 2 + 2

⇒ b = 4.

The parallel line equation is y = 2x + 4.

6).

The line equation is x - y = - 4.

Write the equation in slope - intercept form.

y = x + 4

Comapare the equation with slope - intercept form.

Slope (m) = 1

Because the parallel lines have same slopes, the slope of parallel line through the point (1, - 3) is 1.

Now the parallel line equation is y = x + b.

Find the y - intercept by substituting the point in the parallel line equation say (x, y) = (1, - 3).

- 3  = (1)(1) + b

b = - 3 - 1

⇒ b = - 4.

The parallel line equation is y = x - 4.