Problem 9 - Precalculus Essentials

Question

Use the given conditions to write an equation for each line in point-slope form and general form. Passing through (-2,2) and parallel to the line whose equation is \(2 x-3 y-7=0\)

Step-by-Step Solution

Answer

Equation of the given line parallel to the provided line and passing through the point (-2,2) in General Form is \(2x - 3y + 10 = 0\).

Step 1: Find the slope of the original line

The general form of a line's equation is \(Ax + By + C = 0\). Thus, we can rewrite the given equation \(2x - 3y - 7 = 0\) into the form \(y = mx + c\), which is \(y = \frac{2}{3}x + \frac{7}{3}\). Therefore, the slope of the given line, m, is \frac{2}{3}.

Step 2: Write the equation in point-slope form

Since the new line is parallel to the original line, it has the same slope. The point-slope form of a line is \(y - y1 = m(x - x1)\). Given a point on the line (-2,2), we plug these values into the equation converting it into the form \(y - y1 = m(x - x1)\). Substituting m, x1, y1 with \frac{2}{3}, -2, 2 respectively, we get the equation of the line as \(y - 2 = \frac{2}{3}(x + 2)\).

Step 3: Convert equation into general form

To put this equation into general form, we need to get rid of the fractions by multiplying the equation by 3. So, we get \(3y - 6= 2x + 4\), rearanging gives us: \(2x - 3y + 10 = 0\). This is the final equation of the line in general form.

Key Concepts

Point-Slope Form

The point-slope form is a vital tool in geometry for writing the equation of a line. It is particularly useful when you know a point through which the line passes and the slope of the line. The point-slope form is expressed by the equation \(y - y_1 = m(x - x_1)\). Here, \(m\) represents the slope of the line, and \((x_1, y_1)\) are the coordinates of the given point.

• The equation emphasizes the relationship between a specific point and the slope.
• It allows you to easily substitute known values to find an equation that describes the entire line.

The beauty of the point-slope form is its simplicity and straightforwardness. By just plugging in values of the point and the slope, you can immediately obtain the equation that defines a line.

General Form

The general form of a linear equation is given by \(Ax + By + C = 0\). It's a versatile format because all linear equations can be restructured into this form. This form is particularly useful in algebra because it presents the entire equation without fractions, often with integer coefficients.

• To transition to general form from another form, rearrange terms to ensure all variables and constants are on one side of the equality.
• Typically, multiplication or division might be needed to eliminate fractions, like multiplying by a common denominator.

The general form provides a standardized way of presenting lines and makes it easier to compare different lines and determine properties such as whether two lines are parallel or perpendicular.

Slope of a Line

The slope of a line, denoted by \(m\), depicts its steepness and direction. It's a crucial concept in algebra and geometry, providing insight into how a line behaves on a graph. Slope is calculated as the ratio of the vertical change to the horizontal change between two points on the line, often represented as \(m = \frac{rise}{run}\).

• A positive slope indicates the line rises from left to right.
• A negative slope signifies the line falls from left to right.
• A zero slope characterizes a horizontal line, indicating no vertical change.
• An undefined slope is associated with vertical lines.

The slope is central to understanding line parallels and intersections. Lines that are parallel have identical slopes, which was used in determining the slope in the exercise above. Understanding the slope offers a foundation for deeper exploration of line equations and their graphs.