Question

Let G(u,v)=(2u+v,7u+16v) be a map from the uv-plane to the xy-plane. Find the image of the line v=4u under G in slope–intercept form.

(Use symbolic notation and fractions where needed.)

Answer 1

To find the image of the line v=4u under the map G(u,v)=(2u+v,7u+16v), substitute v=4u into the map equation. The image of the line v=4u under G is the line with the equation y=77x.

Step-by-step explanation:

To find the image of the line v=4u under the map G(u,v)=(2u+v,7u+16v), we substitute v=4u into the map equation:

Let u=x and v=4u, so v=4x. Substituting these values into the map equation, we get:

x-coordinate: 2x+v=2x+4x=6x
y-coordinate: 7x+16v=7x+16(4x)=71x

Therefore, the image of the line v=4u under G is the line with the equation y=71x+6x, which is simplified to y=77x.

Answer 2

The image of the line v = 4u under G is the line
`\(y = (17)/(6)x\)`.

To find the image of the line v = 4u under the transformation G(u,v) = (2u+v, 5u+3v), we need to plug the equation of the line into G.
Given v = 4u, we substitute this into G to get the new coordinates (x, y):
x = 2u + v = 2u + 4u = 6u
y = 5u + 3v = 5u + 3(4u) = 5u + 12u = 17u
Now we express y in terms of x. We already know that x = 6u, so we can solve for u in terms of x:
u = x / 6
Substituting this into the expression we have for y gives:
y = 17u = 17(x / 6)
To put this in slope-intercept form, y = mx + b, where m is the slope and b is the y-intercept, we simplify the expression for y:
y = 17/6 × x + 0
So the slope m is 17/6 and the y-intercept b is 0. The image of the line v = 4u under the transformation G is the line:
y = (17/6)x
This is already in slope-intercept form y = mx + b, where m = 17/6 and b = 0.