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

Question

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

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

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

c. Write the co-ordinates of the intersections of lines M and N with the x-axis.

d. Write the co-ordinates of the intersection of lines M and N with the y-axis.

I have no clue as to how to do this!! Please help!

Answer 1

qa

M : y = 2x + 3

N : y = -x - 6

qb

2x + 3 = -x - 6

3x = -9

x = -3

y = 2(-3) + 3 = -3

answer = (-3,-3)

qc

M : (-3/2,0)

N : (-6,0)

qd

M : (0,3)

N : (0,-6)

Answer 2

The question looks a lot more intimidating than it is. They give you six points for each line, but you only need two points to get the equation of a line. Ignore the other four points.

So, choose any two points from M. I would use (3,9) and (5,13) because they have the smallest numbers, so calculations will be easier.

First find the slope of M:

slope m = (y₂-y₁)/(x₂-x₁)

= (13-9)/(5-3)

= 2

Now find the y-intercept of M:

y-intercept b = y₁ - mx₁

= 9 - 2(3)

= 3

The general equation of a line y = mx + b, so

y = 2x + 3

is the equation for M.

-----

Do the same for N. I would use points (3,-9) and (5,-11).

m = (-11--9)/(5-3)

= -1

b = -9 - (-1)(3)

= -6

Equation for N:

y = -x - 6

-----

M: y = 2x+3

N: y = -x-6

M intersects N when 2x+3 = -x-6.

Solve for x:

2x+3 = -x-6

3x = -9

x = -3

Use x=-3 to find y:

y = 2x+3

= 2(-3) + 3

= -3

M⋂N = (-3, -3)

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A line intersects the x-axis when y=0.

M: Set y to 0 and solve for x:

y = 2x+3

0 = 2x+3

2x = -3

x = -3/2

The x-intercept of M is -3/2. M intersects the x-axis at (-3/2, 0).

Do the same with N:

y = -x-6

0 = -x-6

-x = 6

x = -6

The x-intercept of N is -6.  N intersects the x-axis at (-6, 0).

-----

A line intersects the y-axis when x=0.

You already know that the y-intercept of M is 3, so M intersects the y-axis at (0, 3).

You already know that the y-intercept of N is -6, so N intersects the y-axis at (0, -6).

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