Another math problem? Thank you.?

Question

the lines 2x+y=10 and 7x+8y=53 intersect at A, and the lines 2x-y=12 and x+3y=27 intersect at B. Determine the equation of the line trough A and B. Thank You.

Answer 1

I love these! In a nutshell, solve each pair of simultaneous equations for x and y; then you have two points and you can use y= mx + b to get the line you want--m will just be (y2-y1)/(x2-x1), and then you can pick either x,y pair and plug in to solve for b. So:

solving the first one:

16x + 8y = 80 (first equation multiplied by 8)
7x + 8y = 53
---------------------
9x = 27
X=3, y=4

and for the second pair,

6x -3y=36 (first equation multiplied by 3)
x+3y=27
------------
7x = 63
x=9, y=6

and now,
y=mx+b

m= (6-4)/(9-3) = 2/6 = 1/3

so y=1/3 x + b

and substituting in (9,6) we have

6=1/3 (9) + b
6= 3 + b
b=3

so the equation you want is:

y = 1/3 x + 3


_____________________

I think Pascal forgot a negative sign when solving the first pair of equations..........

Answer 2

First find the points A and B. In both cases, this is done by solving the associated system of equations. For A:

2x+y=10
7x+8y=53

Subtracting 8 times the first equation from the second:

7x+8y - 16x - 8y = 53-80
-9x = -27
x=3

Substituting into the first equation:

2(3) + y = 10
y+6=10
y=4

So point A is (3, 4). Point B is found similarly to be (9, 6) (proof omitted). So then the line will have slope (6-4)/(9-3) = 2/6 = 1/3. The line passes through point A, so in point-slope form it is:

y-4 = 1/3 (x-3)

Placing this into standard form:

3y-12 = x-3
-x+3y = 9

And we are done.

Edit: corrected arithmetical errors in the first post. Specifically, the negative sign that the first poster mentioned I had forgotten.

Answer 3

2x+y = 10
7x+8y =53
-16x -8y =-80
-9x = -27
x = 3 so y= 4 so A(3,4) is 1st point

2x-y=12
x+3y= 27
6x-3y =36
7x = 63
x=9 so y = 6 so B(9,6) is 2nd point
The equation of a line going through two points is:
(y-y1)/x-x1) = (y2-y1)/x2-x1)
(y - 4)/(x- 3) =(6-4)/9-3) = 2/6 = 1/3
y-4 =(1/3)(x-3)
y-4 = x/3 - 1
y = x/3 +3