the mid point of a segment joining (a,b) and (-3,4b) is (2,3a+4).find a & b.

for the x-coordinate:

(a-3)/2 = 2
a-3 = 4
a = 7

for the y-coordinate:
(b + 4b)/2 = 3a + 4
5b = 6a + 8
5b = 42+8 = 50
b = 10

check:
is (2,25) the midpoint of (7,10) and (-3,40) ?
Yes, it is

Oh, I see we've got a math problem here! Let's solve it together, shall we?

To find the midpoint of a segment, we need to average the x-coordinates and the y-coordinates separately.

So, let's equate the x-coordinates:
(a + (-3)) / 2 = 2

Simplifying that, we get:
(a - 3) / 2 = 2

Cross-multiplying, we have:
a - 3 = 4

Adding 3 to both sides:
a = 7

Now, let's equate the y-coordinates:
(b + 4b) / 2 = 3a + 4

Simplifying further:
5b / 2 = 3a + 4

Multiplying both sides by 2, we get:
5b = 6a + 8

Replacing a with 7, we have:
5b = 6(7) + 8
5b = 42 + 8
5b = 50

And finally, dividing by 5:
b = 10

So, after some clown calculations, we find that a = 7 and b = 10.

To find the values of a and b, we can use the midpoint formula.

The midpoint formula states that the midpoint between two points, (x1, y1) and (x2, y2), is calculated as follows:

Midpoint = ( (x1 + x2) / 2, (y1 + y2) / 2 )

In this case, we are given the midpoint as (2, 3a + 4) and the coordinates of two points as (a, b) and (-3, 4b). Let's plug in these values and solve for a and b.

For the x-coordinate:
2 = (a + (-3)) / 2
4 = a - 3
a = 7

For the y-coordinate:
3a + 4 = (b + 4b) / 2
3a + 4 = 5b / 2
3a + 4 = 2.5b
6a + 8 = 5b
6(7) + 8 = 5b
42 + 8 = 5b
50 = 5b
b = 10

Therefore, the values of a and b are a = 7 and b = 10.

To find the values of a and b, we can use the midpoint formula. According to the formula, the coordinates of the midpoint M(xm, ym) of a line segment with endpoints A(x1, y1) and B(x2, y2) can be found using the following equations:

xm = (x1 + x2) / 2
ym = (y1 + y2) / 2

In this case, we have the following information:

Given: Midpoint M(2, 3a+4)

Using the midpoint formula, we can equate the x-coordinates and y-coordinates separately to find a and b.

For x-coordinates:
(x1 + x2) / 2 = 2
(a + (-3)) / 2 = 2
(a - 3) = 4
a = 4 + 3
a = 7

For y-coordinates:
(y1 + y2) / 2 = 3a + 4
(b + 4b) / 2 = 3(7) + 4
(5b) / 2 = 21 + 4
5b / 2 = 25
5b = 50
b = 50 / 5
b = 10

Therefore, the values of a and b are a = 7 and b = 10, respectively.