Question

41\. Write the equation for the line through \((-2,-1)\) that is perpendicular to the line \(y+3=-\frac{2}{3}(x-5)\)

Short answer

The equation of the perpendicular line is \( y = \frac{3}{2}x + 2 \).

Step-by-step solution

Identify the Slope of the Given Line

The given line equation is \( y + 3 = -\frac{2}{3}(x - 5) \). We can rewrite this in the slope-intercept form \( y = mx + b \), where \( m \) is the slope. This gives \( y = -\frac{2}{3}x + \frac{10}{3} - 3 \), simplifying to \( y = -\frac{2}{3}x + \frac{1}{3} \). The slope \( m \) of this line is \(-\frac{2}{3}\).

Determine the Slope of the Perpendicular Line

Perpendicular lines have slopes that are negative reciprocals of each other. Thus, the slope of the line we want to find, which is perpendicular to the given line, is \( \frac{3}{2} \), the negative reciprocal of \(-\frac{2}{3}\).

Write the Equation of the Perpendicular Line

Using the point-slope form of a line \( y - y_1 = m(x - x_1) \), and substituting in the point \((-2, -1)\) and slope \( \frac{3}{2} \), we have: \( y + 1 = \frac{3}{2}(x + 2) \).

Simplify the Equation

Distribute the \( \frac{3}{2} \) on the right side: \( y + 1 = \frac{3}{2}x + 3 \). Subtract \( 1 \) from both sides to solve for \( y \): \( y = \frac{3}{2}x + 2 \).

Key concepts

Equation of a Line

Understanding the equation of a line is crucial for graphing and analyzing linear relationships. In algebra, a line can be represented by different equations, each showing the relationship between the line's slope and intercepts. The standard form is often given as \(ax + by = c\), where \(a\), \(b\), and \(c\) are constants. However, we often use other forms like slope-intercept and point-slope for ease in calculations and interpretations.
These forms allow students to easily plug in values to compare slopes and analyze intersections or parallels between lines. Identifying the equation of a line allows us to understand how two variables in a linear relationship interact.

Slope-Intercept Form

One of the most frequently used forms to write the equation of a line is the slope-intercept form. It is given by \(y = mx + b\). Here, \(m\) represents the slope of the line and determines how steep the line is. The variable \(b\) is the y-intercept, which is where the line crosses the y-axis.
This form is particularly useful when you need to quickly graph a line, as you can start at the y-intercept and use the slope to find another point on the line. Knowing how the components \(m\) and \(b\) affect the line's appearance makes it easy to visualize the line and understand its behavior on a graph.

Point-Slope Form

The point-slope form is versatile and helpful when you know a point on a line and the line's slope. The formula is \(y - y_1 = m(x - x_1)\), where \((x_1, y_1)\) is a specific point on the line, and \(m\) is the slope.
This form is especially advantageous when you're only given a point and a slope, and you need to form the line's equation. The simplicity of substituting directly into the formula makes it a favorite for many problems, like the one where a perpendicular line passes through a given point. Once the point-slope form is set up, transforming it into other forms, like slope-intercept, is straightforward.

Negative Reciprocal

When two lines are perpendicular, their slopes relate specifically through negative reciprocals. This means if one line has a slope \(m\), the slope of the line perpendicular to it will be \(-1/m\). This relationship is key for solving problems involving perpendicularity.
For instance, if a line has a slope of \(-\frac{2}{3}\), a perpendicular line will have the slope \(\frac{3}{2}\). This transformation is crucial when deriving the equation of a line perpendicular to a given line. Understanding the concept of negative reciprocals helps connect the geometry of lines in the coordinate plane with their algebraic expressions, providing deeper insight into the spatial relationships between lines.