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The slope of a line tells you how steep the line is. More formally, it is the ratio that defines how much the y-value of a line increases or decreases as the x-value increases. In mathematical terms, the slope is often represented by the letter 'm.'
You can determine the slope from a linear equation in various forms. For instance, in the slope-intercept form, the slope is simply the coefficient of the x-term. In the exercise, the first equation is given as \(y = \frac{2}{3}x - 1\), so the slope is \(\frac{2}{3}\).
When comparing slopes, remember that:

• Equal slopes mean the lines are parallel.
• Negative reciprocal slopes mean the lines are perpendicular.

To find the slope of an equation given in standard form, like \(2x - 3y = -2\), you must first rearrange it into slope-intercept form \(y = mx + b\). As shown in the solution, the slope is still \(\frac{2}{3}\).

The y-intercept of a line is the point where it crosses the y-axis. This occurs when x is zero. In the slope-intercept form \(y = mx + b\), the y-intercept is represented by 'b.' In the first equation from our example, \(y = \frac{2}{3}x - 1\), the y-intercept is -1.
Knowing the y-intercept is important because it provides a starting point for graphing the line. For the second line initially given in standard form \(2x - 3y = -2\), we converted it to \(y = \frac{2}{3}x + \frac{2}{3}\), revealing a y-intercept of \(\frac{2}{3}\).
These intercepts help understand the placement of the lines in the coordinate plane. However, it’s the slope that primarily determines if lines are parallel.

The slope-intercept form of a linear equation is \(y = mx + b\). This form is a straightforward way to express the equation of a line, where:

• 'm' is the slope.
• 'b' is the y-intercept.

The benefit of the slope-intercept form is that you can quickly identify both the slope and the y-intercept.
For example, in the equation \(y = \frac{2}{3}x - 1\), the slope is \(\frac{2}{3}\) and the y-intercept is -1.
In the exercise's second equation, we converted \(2x - 3y = -2\) into \(y = \frac{2}{3}x + \frac{2}{3}\). Once in this form, it's easy to see that the slope is \(\frac{2}{3}\) and the y-intercept is \(\frac{2}{3}\). Predictably and consistently, the slope-intercept form simplifies the problem of identifying if two lines are parallel or not.

The standard form of a linear equation is given as \(Ax + By = C\), where A, B, and C are integers, and A should be non-negative. This form does not immediately reveal the slope and y-intercept, requiring some manipulation.
To find the slope and y-intercept from standard form, convert it to slope-intercept form \(y = mx + b\). This requires solving for 'y.'
As demonstrated, converting the equation \(2x - 3y = -2\) involves:

• Subtracting 2x from both sides to get \(-3y = -2x - 2\).
• Dividing every term by -3 to result in \(y = \frac{2}{3}x + \frac{2}{3}\).

Now, the slope, \(\frac{2}{3}\), and the y-intercept, \(\frac{2}{3}\), are clear.
Understanding how to transition between these forms is crucial for solving linear equations and determining relationships like parallelism.