If is a line whose equation is y = 2x- 1, find the equation of the image of under each of the following translations:(x, y) S (x +3, y)

To find the equation of the image of a line after a translation by (x + 3, y), we need to shift every point on the line by 3 units to the right.

For the given line with equation y = 2x - 1, we can rewrite it as:

y - 2x = -1

Now, let's apply the translation by adding 3 to the x-coordinate:

y - 2(x + 3) = -1

Expanding the equation, we get:

y - 2x - 6 = -1

Simplifying further, we have:

y - 2x = 5

Therefore, the equation of the image of the line after the translation (x, y) S (x + 3, y) is y - 2x = 5.

To find the equation of the image of a line under a translation, we need to apply the translation to every point on the line and then determine the equation of the new line.

Let's start with the original line equation: y = 2x - 1.

The given translation is (x, y) → (x + 3, y). This translation shifts every x-coordinate 3 units to the right while keeping the y-coordinate unchanged.

To apply this translation to the original line equation, we can substitute x + 3 for x in the equation.

Replacing x with x + 3, the new equation becomes:
y = 2(x + 3) - 1

Simplifying:
y = 2x + 6 - 1
y = 2x + 5

Therefore, the equation of the image of the line y = 2x - 1 under the translation (x, y) → (x + 3, y) is y = 2x + 5.

let (x,y) ---> (X,Y) where (X,Y) = (x+3,y)

so X = x+3
x = X-3 and y = Y

new equation:
Y = 2(X-3) - 1
Y = 2X - 7

or in the normal way : y = 2x - 7

test: take any two points on the original line, say
(0,-1) and (2,3)
after the translation
(0,-1) ---> (3,-1)
(2.3) ----> (5,3)

Do these new points lie on my new equation.
A quick mental check shows they are