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The slope-intercept form is one of the most popular ways to express a linear equation. It is given by the formula \( y = mx + c \), where \( m \) represents the slope of the line, and \( c \) represents the y-intercept. Writing an equation in this form helps to easily identify these two key components of a line.
To convert the given equation \( 4x + 6y = 12 \) into the slope-intercept form, we rearrange the equation so that \( y \) is isolated on one side. Subtract \( 4x \) from both sides to get \( 6y = -4x + 12 \).
Next, divide each term by 6, as we need to solve for \( y \). This simplifies to \( y = -\frac{2}{3}x + 2 \). Now, the equation is clearly in slope-intercept form, which greatly aids the process of graphing a line.

Knowing how to find the slope and y-intercept from a linear equation is crucial. Once an equation is expressed in the slope-intercept form \( y = mx + c \), identifying the slope and y-intercept becomes straightforward. In our equation, \( y = -\frac{2}{3}x + 2 \), the coefficient of \( x \) \(-\frac{2}{3}\) is the slope \( m \), and 2 is the y-intercept \( c \).
The slope \( m \) of \(-\frac{2}{3}\) tells us that the line declines by 2 units vertically for every 3 units it moves horizontally. It signifies the steepness and direction of the line.
The y-intercept is the point where the line crosses the y-axis, which in this case is \( (0, 2) \). This point is vital for starting your plot as it provides a fixed reference to draw your line.

Plotting a graph using the slope-intercept form involves a few clear steps. Start by plotting the y-intercept, which is a pivotal point on the graph. For our equation, the y-intercept is at \( (0, 2) \). Mark this point clearly on the y-axis.
Next, utilize the slope \( m = -\frac{2}{3} \). The negative sign indicates that the line will slant downwards. From the y-intercept \( (0, 2) \), move down 2 units (a reflection of the numerator) and then 3 units to the right (the denominator's influence). This brings you to the next point at \( (3, 0) \).
With these two points, you already have a basic guide for the graph. Simply draw a line that passes through the y-intercept and this new point. Extending this line will give the full representation of the linear equation \( 4x + 6y = 12 \). Check your graph by verifying it intersects at the y-intercept and follows the correct slope.