Question

Write the equation in the slope-intercept form, and then find the slope and \(y\) -intercept of the corresponding lines. $$ 2 x-3 y-12=0 $$

Short answer

The given equation is \(2x - 3y - 12 = 0\). To rewrite it in slope-intercept form, we get \(y = \frac{2}{3}x - 4\). The slope (m) is \(\frac{2}{3}\) and the y-intercept (b) is -4.

Step-by-step solution

Rewrite the given equation in slope-intercept form

To rewrite the given equation in slope-intercept form (y = mx + b), first, isolate y by getting it alone on one side of the equation. The given equation is: $$2x - 3y - 12 = 0$$ Add 3y to both sides of the equation, and then subtract 2x from both sides: $$3y = 2x - 12$$ Now, divide both sides by 3 to isolate y: $$y = \frac{2}{3}x - 4$$

Identify the slope and y-intercept

Now that the equation is in slope-intercept form, we can easily identify the slope and y-intercept directly from the equation: $$y = \frac{2}{3}x - 4$$ Here, the slope (m) is the coefficient of x, which is \(\frac{2}{3}\), and the y-intercept (b) is the constant term, which is -4. So, the slope is \(\boxed{\frac{2}{3}}\) and the y-intercept is \(\boxed{-4}\).