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Understanding how to find the slope of a linear equation is a fundamental algebraic skill. The slope measures the steepness of a line and is usually represented by the letter 'm' in equations. Remember that the slope is essentially the rate at which the 'y' value changes with respect to a change in the 'x' value.

Using the example provided, the slope is found after rewriting the given equation into the slope-intercept form, which is an expression of the equation as \(y = mx + b\). Here's how you can approach it:

• Isolate the 'y' variable on one side of the equation.
• Identify the coefficient (number in front) of 'x', after the equation is in the form \(y = mx + b\).

Once the equation is in slope-intercept form, the slope is simply the coefficient of 'x'. In the exercise, dividing each term by 3 transforms the equation to \(y = 2 + \frac{2}{3}x\). Here, the slope (m) is \(\frac{2}{3}\).

The y-intercept of a line is the point where the line crosses the y-axis on a graph. This value is significant because it gives us a starting point for graphing the linear equation and is represented by the letter 'b' in the slope-intercept form \(y = mx + b\).

To find the y-intercept from a linear equation, one must follow these steps:

• Rewrite the given equation in slope-intercept form, isolating 'y' on one side.
• Identify the constant term, which is the value left without an 'x' next to it after simplifying the equation.

In our exercise, after dividing each term by 3 to get \(y = 2 + \frac{2}{3}x\), the constant term 2 signifies the y-intercept. As a result, the y-intercept, 'b', is 2, indicating the graph of this function crosses the y-axis at the point (0,2).

Linear equations form straight lines when graphed on a coordinate plane. These equations have one or more variables, but every term is either a constant or the product of a constant and a single variable.

A standard linear equation takes the form of \(Ax + By = C\), where 'A', 'B', and 'C' are constants. However, to graph or analyze such equations, it's often helpful to manipulate them into slope-intercept form \(y = mx + b\), as it visually displays both the slope 'm' and the y-intercept 'b' of the linear function. For our exercise, the original equation \(3y = 6 + 2x\) is a linear equation with variable coefficients for 'y' and 'x'. Upon algebraic manipulation, it assumes the slope-intercept form, clarifying the rate of change (slope) and the initial value (y-intercept) for graphing purposes.

Algebraic manipulation involves rearranging and simplifying equations to solve for unknown variables or to put the equation into a more useful form. It uses operations such as adding, subtracting, multiplying, dividing, and factoring.

In the context of our exercise, the equation \(3y = 6 + 2x\) had to undergo algebraic manipulation to find the slope and y-intercept. This process involves:

• Dividing each term by 3 to isolate 'y' on one side of the equation, resulting in \(y = 2 + \frac{2}{3}x\).
• Recognizing that the coefficient of 'x' and the standalone constant represent the slope and y-intercept, respectively.

This type of manipulation is essential for analyzing and graphing linear equations and forms the basis for much of algebra.