Question

Find the equation of the line parallel to x+3y = 4 and passing through the point (2, 5)

Answer 1

`\displaystyle y-5=-(1)/(3)(x-2)`

Explanation:

Equation of a Line

Two lines are parallel if they have the same slope. We are given the equation of the line:

x+3y=4

To find the slope of this line, we must solve for y:

`\displaystyle y=-(1)/(3)x+(4)/(3)`

The slope of this line is:

`\displaystyle -(1)/(3)`

The equation of the new line has the same slope. To find its equation, we use the point-slope form of the line:

y-k=m(x-h)

Where (h,k)=(2,5). Thus:

`\boxed{\displaystyle y-5=-(1)/(3)(x-2)}`

Answer 2

The equation of the parallel line is x + 3y = 17

Explanation:

Parallel lines have the same slopes and different y-intercept

The form of the equation y = m x + b, where

• m is the slope of the line

• b is the y-intercept

∵ The equation of the given line is x + 3y = 4

→ We must put it in the form above to find its slope

→ Subtract x from both sides to move x to the right side

∵ x - x + 3y = 4 - x

∴ 3y = 4 - x

→ Divide both sides by 3 to make the coefficient of y = 1

∴
`(3y)/(3)=(4)/(3)-(1)/(3)x`

∴ y =
`(4)/(3)` -
`(1)/(3)` x

→ Compare it with the form of the equation above

∴ m =
`-(1)/(3)`

∵ Parallel lines have the same slopes

∴ The slope of the parallel line is
`-(1)/(3)`

→ Put it in the form of the equation

∴ y =
`-(1)/(3)` x + b

→ To find b substitute x and y in the equation by the coordinates

of a point on the line

∵ The line passes through the point (2, 5)

∴ x = 2 and y = 5

→ Substitute them in the equation

∴ 5 =
`-(1)/(3)` (2) + b

∴ 5 =
`-(2)/(3)` + b

→ Add
`(2)/(3)` to both sides

∴
`(17)/(3)` = b

→ Substitute it in the form of the equation above

∴ y =
`-(1)/(3)` x +
`(17)/(3)`

→ Multiply each term by 3

∴ 3y = - x + 17

→ Add x to both sides

∴ x + 3y = 17

∴ The equation of the parallel line is x + 3y = 17