Can someone help with graphing, Please?

Question

The points (3, 9), (5, 13), (15, 33), (34, 71), (678, 1359), and (1234, 2471) all lie on line M.

The points (3, -9), (5, -11), (15, -21), (34, -40), (678, -684), and (1234, -1240) all lie on line N.

Form the equations of both the lines. Show your work.

What are the co-ordinates of the point of intersection of lines M and N?

Answer 1

Well, since you know that there are two lines, you can therefore say that there are two linear equations involved.

Simply pick any two points for each line (since two points determine a line) and derive the equation using the two-point form of the linear equation: y-y1=(y2-y1/x2-x1)(x-x1)

line_1

(3,9) and (5,13)

y-9 = [(13-9)/(5-3)](x-3)

y = 2(x-3) + 9

y = 2x + 3

line_2

(3,-9) and (5,-11)

y+9 = [(-11+9)/(5-3)](x-3)

y = -1(x-3) - 9

y = -x - 6

You can then graph the two lines using their slope-intercept form equations (y=mx+b) and determine their point of intersection.

Alternatively, you can solve this system of linear equations:

y = 2x + 3

y = -x - 6

By transitivity: 2x + 3 = -x -6

Solve for x:

3x = -9

x = -3

Solve for y using x:

y = 2(-3) + 3

y = -3

.: (-3,-3) is the point of intersection of the two lines.

Answer 2

You just need 2 points since they all lied in a line...

y - 9 = {(13 - 9) / (5 - 3)} (x - 3)

y - 9 = (2) (x-3)

y - 9 = 2x - 6

the equation for line M is 2x - y = -3

y - (-9) = [{-11 - (-9)} / {5 - 3}] (x - 3)

y + 9 = [ -2 / 2 ] (x - 3)

y + 9 = [-1] (x - 3)

y + 9 = -x + 3

the equation for line N is x + y = -6

There are two ways to find their point of intersection:

===> Substitution Method or Elimination Method...

2x - y = -3    --->1st Eq.

x + y = -6    ---> 2nd Eq.

For substitution method...

Get the value x or y using the eq.

You can use the 1st or 2nd eq.

I use the 2nd eq.

x = -y -6

Substitute this value of x to other equation which is the 1st eq.

2(-y -6) - y = -3

-2y -12 -y = -3

-3y = 9

y = -3

Now you have the value for y, then use it again in the eq - 1st or 2nd

I use the 1st eq.

2x - (-3) = -3

2x + 3 = -3

2x = -6

x = -3

The point of intersection is (-3,-3)

You can also use the elimination method...

2x - y = -3

x+ y = -6

_________

3x = -9

x = -3

I add the two equation to eliminate y... So we have now the value of x which is -3...

Now you have the value of x... Do substitution to 1st or 2nd...

You will get the same point of intersection...

Equation for line M... 2x - y = -3

Equation for line N... x + y = -6

Point of Intersection.... (-3, -3)

That's it... GIVE ME MY 10 points!