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The slope-intercept form of a linear equation is one of the most commonly used forms because it provides key information about the line at a glance.
This form is written as:
= y = mx + b
In this equation:

• y represents the dependent variable
• x represents the independent variable
• m is the slope of the line
• b is the y-intercept, which is the point where the line crosses the y-axis

The slope-intercept form is very useful for quickly graphing a line.

• The slope m tells you how steep the line is.
• It's the ratio of the change in y to the change in x, often described as 'rise over run'.
• The y-intercept b gives an exact starting point on the y-axis.

In our exercise, after converting the given line equation 2x - 3y = 4 to slope-intercept form, we identified the slope as m = 2/3.

Another helpful form for linear equations is the point-slope form. This form is especially useful when you know the slope of a line and a point it passes through.

The point-slope form is written as:
\[ y - y_1 = m (x - x_1) \]
In this equation:

• m represents the slope of the line
• (x_1, y_1) is a point through which the line passes

To use this form, just substitute the specific point's coordinates and the slope into the equation.

For this exercise, where our given point was (-3, 7), and slope m = 2/3, we substituted these values into the point-slope formula:
\[ y - 7 = \frac{2}{3} (x + 3) \]
Then, it's just a matter of simplifying the equation further to attain the slope-intercept form again, which for this problem resulted in:
\[ y = \frac{2}{3} x + 9 \]

Understanding when lines are parallel is crucial in linear algebra.

Parallel lines share a unique property—they have the exact same slope but different y-intercepts.

For two lines to be parallel, their slopes must be identical. If the equation of a line is in slope-intercept form y = mx + b, any line parallel to it will have an equation of the form y = mx + c, where m is the same for both lines, but b and c are different.

In this exercise, our given line was originally in the form 2x - 3y = 4.

After converting it to slope-intercept form, we got y = (2/3)x - 4/3.

The slope here is 2/3.

Thus, any line parallel to this must also have a slope of 2/3.

We used the slope 2/3 and the point (-3, 7) in the point-slope form equation:
\[ y - 7 = \frac{2}{3} (x + 3) \]
Simplifying this gave us the final equation of the parallel line:
\[ y = \frac{2}{3} x + 9 \]
Both lines have the same slope, 2/3, ensuring they are parallel.