The found value are-
- Equation of the translated line: y = mx − am + 2b.
- The value of the y -intercept: 2b − am.
<h3>What is general form of straight line?</h3>
A straight line's general equation is y = mx + c, in which m is the gradient and y = c is the value at which the line intersects the y-axis. This number c is known as the y-axis intercept.
Now, as per the given question;
The general equation of the line is given as;
y = mx + b
Each point (x,y) of the line is moved to the point (x+a, y+b) by the translation:
(x,y) ---> (x+a, y+b)
We find the original line's x and y intercepts:
Put y=0 in general equation of the line
⇒mx+b = 0
⇒x = −b/m
When, x=0
⇒y = m⋅0 + b
⇒ y = b
A, (0,b) and B( -b/m, 0)
We find the translational coordinates of the points A,B:
A(0,b) ---> A′ (0+a,b+b)
A(0,b) ---> A′ (a,2b)
And,
B( -b/m, 0) ----> B'( -b/m + a, 0 + b)
B( -b/m, 0) ----> B'( -b/m + a, b)
To find the equation of the line's translation, we use the points A and B ′:
m' = (2b - b)/[a - (-b/m +a)]
m' = b/(b/m)
m' =m
And,
y−yA' = m' (x−xA')
y−2b=m(x−a)
y=mx−am+2b
The translation of the line's x and y intercepts are:
Put x=0
⇒y = m⋅0−am+2b
⇒y = 2b−am
And, put y=0
⇒0 = mx−am+2b
⇒x = a - (2b/m)
Therefore, the equation of the translate line is found to be y = mx − am + 2b.
And, the value of the y-intercept is found to be 2b−am.
To know more about the general form of straight line, here
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