a We are asked to select two equations which form a system of equations that has infinitely many solutions.

For a system to have infinitely many solutions, it must mean that the lines intersect at infinitely many points. Such lines are said to be coincidental, and they have the same slope and $y -$ intercept.

First, we will rewrite each equation in slope-intercept form and find the slope and the

y -

intercept for each equation. Let's do it for

3 x + y = 12!

3 x + y = 12

$LHS - 3 x = RHS - 3 x$

y = - 3 x + 12

Therefore, the slope is equal

- 3

and the

y -

intercept is

12 .

Let's do this for the remainder of the given equations, too.

Equation	Slope-Intercept Form	Slope	$y -$ intercept
$3 x + y = 12$	$y = - 3 x + 12$	$- 3$	$12$
$3 x + 2 y = 12$	$y = - \frac{3}{2} x + 6$	$- \frac{3}{2}$	$6$
$6 x + 2 y = 6$	$y = - 3 x + 3$	$- 3$	$3$
$3 y + 9 x = 36$	$y = - 3 x + 12$	$- 3$	$12$
$2 y - 6 x = 12$	$y = 3 x + 6$	$3$	$6$
$9 x - 3 y = - 18$	$y = 3 x + 6$	$3$	$6$

From the table we can see that

3 x + y = 12

and

3 y + 9 x = 36

have the same slope and

y -

intercept. Therefore, the following system of equations has infinitely many solutions.

{3 x + y = 12 3 y + 9 x = 36

Exercise