Rewriting the Equation 6x - 4y 24 in Slope-Intercept Form: A Comprehensive Guide

Algebra is a fundamental part of mathematics, and understanding how to manipulate and solve equations is crucial. One common task in algebra is to rewrite linear equations in the slope-intercept form, which is (y mx b). This form is particularly useful for graphing and understanding the properties of the line.

Step-by-Step Guide to Rewriting the Equation

The given equation is:

6x - 4y 24

To convert this equation into slope-intercept form ((y mx b)), we need to follow a series of algebraic steps. Let's break it down:

Step 1: Isolate y

The first step is to isolate the (y) term on one side of the equation. We start by moving the (6x) term to the right side:

6x - 4y 24

-4y -6x 24

Step 2: Divide by the Coefficient of y

Next, we want the (y) term to be by itself, so we divide every term by (-4):

y frac{-6x 24}{-4}

y frac{-6x}{-4} frac{24}{-4}

y frac{3}{2}x - 6

Step 3: Simplify the Fractions

In the final step, we simplify the fractions. Note that (-frac{6x}{-4}) simplifies to (frac{3x}{2}), and (frac{24}{-4}) simplifies to (-6):

y frac{3}{2}x - 6

Understanding the Slope-Intercept Form

In the slope-intercept form, (y mx b), the coefficient of (x) (in this case, (frac{3}{2})) represents the slope ((m)) of the line, and the constant term (in this case, (-6)) represents the y-intercept ((b)).

Slope and Y-Intercept

The slope ((m)) of the line is (frac{3}{2}). This means that for every increase of 1 unit in (x), (y) increases by (frac{3}{2}) units. The y-intercept ((b)) is (-6), which is the value of (y) when (x 0).

Alternative Methods for Rewriting the Equation

There are various methods you can use to rewrite the equation. Here are a couple of alternative approaches:

Method 1: Using Division Directly

You can also divide the entire equation by 4 directly from the beginning:

6x - 4y 24

-4y -6x 24

y -frac{6x}{4} frac{24}{4}

y -frac{3}{2}x 6

Method 2: Narrowing Down to Essential Steps

Another straightforward approach is to subtract (6x) from both sides and then divide by 4:

6x - 4y 24

-4y -6x 24

y frac{-6x 24}{-4}

y frac{-6x}{-4} frac{24}{-4}

y frac{3}{2}x - 6

Conclusion

Mastering the process of converting equations into slope-intercept form is a valuable skill in algebra. By following the steps outlined above, you can easily transform linear equations into a form that is both insightful and practical for graphing and analysis. Whether you choose to isolate (y) first or divide directly, the methods will always lead you to the same result.