Question

Line L is mapped onto line M by a dilation centered at the origin with a scale factor of 2 the equation of line L is 3x - y = 4 determine and state an equation for line M

Answer 1

Equation of the line M is
`2y=6x-8` i.e.
`y=3x-4`

Explanation:

We are given that,

Line L is mapped onto the line M by a dilation at the origin by a scale factor of 2.

Now, the equation of the line L is
`3x-y=4` i.e.
`y=3x-4`.

As, the scale factor of dilation is 2.

Also, dilation only changes the size of the figure and have no other impact.

Thus, the equation of M will be same as that of L but increased in size.

The equation of line M will be
`2y=2(3x-4)` i.e.
`2y=6x-8`.

Hence, equation of the line M is
`2y=6x-8`.

Answer 2

Answer: 3x - y = 8

Explanation:

Since, if a line having points represented by (x,y) is dilated about origin with a scale factor k,

Then the coordinates of the dilated line is obtained by the rule,

`(x,y)\rightarrow (kx,ky)`

Here, line L is,

3x - y = 4

The x-intercept and y-intercept of line L are (4/3,0) and (0,-4) respectively,

If the line L is mapped onto line M by a dilation centered at the origin with a scale factor of 2,

Then by the above definition,

The points of the line M are,

(2×4/3, 2×0) and (2×0, 2×-4) ⇒ (8/3,0) and (0,-8)

Hence, the equation of line M passes through the points (8/3,0) and (0,-8)

`y+0=(-8-0)/(0-8/3) (x-(8)/(3))`

`\implies y = 3((3x-8)/(3))`

`\implies 3y=3(3x-8)`

`\implies y = 3x - 8`

`\implies 3x - y = 8`