Question

How many solutions does the system of equation have?
4x + 16y = 12

y = -1/4x + 9/12

A. one
B. two
C. infinitely many
D. none

Answer 1

We can determine the number of solutions from the slope intercept form of the equations.

Slope intercept form of equation 1 is:

$y=- \frac{1}{4}x+3/4$

Slope intercept form of second equation is:

$y=- \frac{1}{4}x+ \frac{9}{12} \\ \\ y= - \frac{1}{4}x+ \frac{3}{4}$

The slope and the y-intercept of both equations are the same. This means, the two lines are lying over each other and hence they infinite number of solutions.

So, the correct answer is option C

Answer 2

The equation no solution. This is because the lines are parallel.
The general equaltion of a line is,

Y=mX+c, where m is the gradient and c is the Y-intercept.

For the first equation,
4x + 16y = 12
16y=-4x+12
y=(-1/4)x+12/16
gradient =-1/4

For the second equation, Y = (-1/4)x +9/12,  the gradient = -1/4
 If the gradients are equation then, the lines do not meet hence no solutions.