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The slope-intercept form of a linear equation is a way to express the equation of a line using its slope and y-intercept. It is generally written as \( y = mx + b \). Here, \( m \) is the slope of the line, which shows how steep the line is, and \( b \) is the y-intercept, which indicates where the line crosses the y-axis. For example, in the equation \( y = \frac{3}{4}x + 2 \), the slope \( m \) is \( \frac{3}{4} \) and the y-intercept \( b \) is 2. This equation tells us that for every 4 units we move horizontally, the line moves 3 units vertically. This form is particularly useful when you need to quickly sketch a graph or determine if two lines are parallel (by comparing their slopes). Additionally, slope-intercept form makes it simple to predict how changing \( m \) or \( b \) would transform the line on a graph.

The point-slope form is another useful way to write the equation of a line. It is most helpful when you know a point on the line and the slope but not the y-intercept. The form is given by \( y - y_1 = m(x - x_1) \), where \( (x_1, y_1) \) is a known point and \( m \) is the slope of the line. This form allows direct input of these known values to establish an equation for the line. For the exercise, since the slope \( m \) is known to be \( \frac{3}{4} \) and the point \( (2, -3) \) is given, we can plug these into the point-slope form: * \( y + 3 = \frac{3}{4}(x - 2) \). By solving for \( y \), it becomes easier to convert this equation into the slope-intercept form, where it is often simpler to read and interpret the graphically.

Parallel lines in a plane are lines that never intersect. This means they have the same slope, but different y-intercepts. The fact that parallel lines share the same slope is really crucial, as it simplifies finding equations for lines parallel to a given line. In our exercise, we were asked to find a line parallel to \( y = \frac{3}{4}x + 2 \). Knowing that parallel lines share the same slope, our new line must also have a slope of \( \frac{3}{4} \). By applying the point-slope form using the point \( (2, -3) \) and this slope, we establish the equation of our new line as \( y = \frac{3}{4}x - \frac{9}{2} \). The difference in the y-intercept from the original line is what keeps these two lines from intersecting, confirming their parallel nature.