If the lines 2x+3y+12=0 and y=mx+1 meet at a point whose coordinates are equal, what must m equal?

Question

I really need help with this I have no idea what to do

Answer 1

If it means x = y at the intersection point, I suggest replacing y with x in both equations, solving the first equation for x, substituting the value of x for both x and y in the second equation, and solving the second equation for m.

 

First equation:

 

2x + 3y + 12 = 0

2x + 3x + 12 = 0

5x = -12

x = -12/5

 

Second equation:

 

y = mx + 1

x = mx + 1

m = (x - 1)/x = (-12/5 - 1)/(-12/5) = (-17/5)/(-12/5) = 17/12

 

m = 17/12

 

(x,y) = (-12/5,-12/5)

 

Now plug these values of x and y into the first equation and test to see it it is satisfied.  Plug them into the second equation, and plug in the value of m, and see if the second equation is also satisfied.

 

2x + 3y + 12 = 2(-12/5) + 3(-12/5) + 12 = -60/5 + 12 = -12 + 12 = 0  [good]

mx + 1 = (17/12)(-12/5) + 1 = -17/5 + 5/5 = -12/5  [good]