Question

1. $6y - 12x = 4$ $3y = 6x + 2$

$m = $ ____ $m = $ ____
$b = $ ____ $b = $ ____
circle: one, none, infinite

Explanation:

Step1: Rewrite first equation in slope - intercept form ($y = mx + b$)

Start with \(6y-12x = 4\). Add \(12x\) to both sides: \(6y=12x + 4\). Then divide each term by 6: \(y=\frac{12x}{6}+\frac{4}{6}\), which simplifies to \(y = 2x+\frac{2}{3}\). So for the first equation, \(m = 2\) and \(b=\frac{2}{3}\).

Step2: Rewrite second equation in slope - intercept form

Start with \(3y=6x + 2\). Divide each term by 3: \(y=\frac{6x}{3}+\frac{2}{3}\), which simplifies to \(y = 2x+\frac{2}{3}\). So for the second equation, \(m = 2\) and \(b=\frac{2}{3}\).

Step3: Determine the number of solutions

Since both equations have the same slope (\(m = 2\)) and the same y - intercept (\(b=\frac{2}{3}\)), the two lines are coincident (they are the same line). So there are infinite solutions.

Answer:

For \(6y - 12x=4\): \(m = 2\), \(b=\frac{2}{3}\)
For \(3y=6x + 2\): \(m = 2\), \(b=\frac{2}{3}\)
Number of solutions: infinite