Understanding the Equation
The equation given is 2x - 3y - 12 = 0. This is a linear equation in two variables, x and y. It can be rearranged to express y in terms of x or vice versa.
Rearranging the Equation
To find y in terms of x, we can rearrange the equation:
- Start by isolating the term with y:
3y = 2x - 12
- Next, divide every term by 3:
y = (2/3)x - 4
Finding the Slope and Intercept
The equation y = (2/3)x - 4 is now in slope-intercept form (y = mx + b), where:
- Slope (m): 2/3, which indicates the steepness of the line.
- Y-intercept (b): -4, meaning the line crosses the y-axis at (0, -4).
Graphing the Equation
To graph the equation, follow these steps:
- Start at the y-intercept (0, -4).
- Use the slope to find another point: from (0, -4), move up 2 units and right 3 units to reach (3, -2).
- Plot this point and draw a straight line through both points.
Finding Solutions
Every point (x, y) on this line is a solution to the equation 2x - 3y - 12 = 0. For example:
- If x = 0: y = -4 (point: (0, -4))
- If x = 3: y = -2 (point: (3, -2))
Conclusion
The solution set for the equation consists of all points on the line defined by y = (2/3)x - 4. Each point represents a unique (x, y) pair that satisfies the equation.