3x Y 6 In Slope Intercept Form

The equation 3x + y = 6 is a linear equation in two variables that describes a straight line in the coordinate plane. In many algebraic and geometric contexts, it is convenient to rewrite such an equation in slope‑intercept form, because this form immediately reveals the line’s slope and y‑intercept. The slope‑intercept representation is written as

This is where a lot of people lose the thread.

[ y = mx + b ]

where (m) denotes the slope and (b) the y‑intercept. By isolating (y) on one side of the equation, we can read off these two crucial parameters without further computation. This article explains how to convert (3x + y = 6) into slope‑intercept form, interprets the resulting values, and explores practical uses of this representation.

Understanding Slope‑Intercept Form

Definition and Components

The slope‑intercept form (y = mx + b) is a standard way to express a linear equation.

• (m) is the slope, which measures the steepness and direction of the line. A positive slope rises from left to right, while a negative slope falls.
• (b) is the y‑intercept, the point where the line crosses the y‑axis. It is the value of (y) when (x = 0).

Because the form isolates (y), it provides a direct method for graphing: start at the y‑intercept ((0, b)) and then move according to the slope (m) (rise over run).

Why Convert to Slope‑Intercept Form?

Converting an equation to this form simplifies several tasks:

1. Graphing – The slope and y‑intercept give a clear starting point and direction.
2. Comparison – Two lines can be compared by examining their slopes and intercepts.
3. Solving Systems – When equations are in slope‑intercept form, substitution becomes straightforward.
4. Interpretation – In real‑world contexts, the slope often represents a rate of change, and the intercept represents an initial value.

Converting (3x + y = 6) to Slope‑Intercept Form

Step‑by‑Step Algebraic Manipulation

To isolate (y), we perform elementary operations that keep the equation balanced.

1. Subtract (3x) from both sides
   [ 3x + y - 3x = 6 - 3x ]
   Simplifying gives
   [ y = -3x + 6 ]

2. Identify the slope and y‑intercept
   Comparing (y = -3x + 6) with the template (y = mx + b), we see

   • (m = -3)
   • (b = 6)

Thus, the slope‑intercept form of the original equation is

[ \boxed{y = -3x + 6} ]

Verification

A quick check confirms the transformation:

• When (x = 0), (y = -3(0) + 6 = 6), matching the y‑intercept.
• When (x = 1), (y = -3(1) + 6 = 3); the change in (y) is (-3) for a change in (x)