Elena writes the equation $6x + 2y = 12$. We need to write a new equation that has: a. exactly one solution in common with Elena's equation b. no solutions in common with Elena's equation c. infinitely many solutions in common with Elena's equation

1. Problem Description

Elena writes the equation $6x + 2y = 12$. We need to write a new equation that has:

a. exactly one solution in common with Elena's equation

b. no solutions in common with Elena's equation

c. infinitely many solutions in common with Elena's equation

2. Solution Steps

a. Exactly one solution in common:

To have exactly one solution in common, the lines must intersect at a single point. This means the lines must have different slopes.

The given equation is $6x + 2y = 12$. We can rewrite it in slope-intercept form ($y = mx + b$) as follows:

$2y = -6x + 12$

$y = -3x + 6$

The slope of Elena's equation is $-3$. To get exactly one solution, we need a different slope. Let's pick a slope of $0$. An example equation would be $y=0$, which means $0x + 1y = 0$. Substituting $y=0$ into $6x + 2y = 12$, we get $6x + 2(0) = 12$, which gives $6x=12$, so $x=2$. Thus, the solution (2, 0) satisfies both equations.

b. No solutions in common:

To have no solutions in common, the lines must be parallel and distinct. This means they must have the same slope but different y-intercepts.

The original equation is $y = -3x + 6$. We need to keep the same slope ($-3$) but change the y-intercept. Let's change the y-intercept to $7$. Then the equation is $y = -3x + 7$. Multiplying by $2$ and rearranging, we get $2y = -6x + 14$, so $6x + 2y = 14$.

c. Infinitely many solutions in common:

To have infinitely many solutions, the two equations must represent the same line. This means we can multiply Elena's equation by a constant.

Elena's equation is $6x + 2y = 12$. If we multiply this equation by, for example, $2$, we get $12x + 4y = 24$.

3. Final Answer

a. An equation with exactly one solution in common is $y = 0$.

b. An equation with no solutions in common is $6x + 2y = 14$.

c. An equation with infinitely many solutions in common is $12x + 4y = 24$.