Understanding the Slope of 3x - 2y 12: Converting to Slope Intercept Form

Linear equations are a fundamental part of algebra and are often encountered in various fields, from basic mathematics to more complex applications in science and engineering. Understanding the slope of a linear equation helps us understand the behavior and direction of the line represented by the equation. In this article, we will explore how to convert a linear equation to its slope intercept form and specifically focus on the equation 3x - 2y 12. We'll delve into the process of finding the slope and the y-intercept, making the equation more accessible for visual and analytical purposes.

Converting Linear Equations to Slope Intercept Form

The slope intercept form of a linear equation is given by y mx b. In this form, m represents the slope of the line, and b is the y-intercept, which is the point where the line crosses the y-axis.

Step-by-Step Conversion

1. Start with the given equation: 3x - 2y 12 2. Isolate the y term on one side of the equation. To do this, subtract 3x from both sides:

3x - 2y  12

-2y  -3x   12

3. To solve for y, divide every term by -2:

y  1.5x - 6

Now the equation is in the slope intercept form, where the slope (m) is 1.5 and the y-intercept (b) is -6.

Interpreting the Slope and Y-Intercept

The slope (1.5) indicates the rate of change of y with respect to x. In simpler terms, for every unit increase in x, y increases by 1.5 units. The y-intercept (-6) is the value of y when x is 0. This is where the line crosses the y-axis.

Visualizing the Line

With the equation in slope intercept form, we can easily plot the line on a coordinate system. Starting from the y-intercept (-6), we can move 1 unit to the right and 1.5 units up to find the next point on the line. This process can be repeated to plot more points, which can then be connected to form a straight line.

Applications of Slope Intercept Form

The slope intercept form has numerous practical applications. It is used in economics to model linear relationships between variables, in physics to represent linear motion, and in engineering to analyze linear structures and systems. By understanding how to work with this form, one can effectively interpret and apply linear equations in real-world scenarios.

Keywords: slope, slope intercept form, linear equations