Problem 41 - College Algebra with Modeling and Visualization

Question

Find the slope-intercept form for the line satisfying the conditions. Perpendicular to \(y=-2 x,\) passing through \((-2,5)\)

Step-by-Step Solution

Answer

The equation is \(y = \frac{1}{2}x + 6\).

Step 1: Identify the Slope of the Given Line

The given line equation is in the form of \(y = mx + c\), where \(m\) is the slope. Here, the equation is \(y = -2x\), so the slope \(m\) is \(-2\).

Step 2: Find the Slope of the Perpendicular Line

Lines that are perpendicular to each other have slopes that are negative reciprocals. The negative reciprocal of \(-2\) is \(\frac{1}{2}\). Thus, the slope of the line we need to find is \(\frac{1}{2}\).

Step 3: Write the Point-Slope Form Equation

Using the point-slope form equation \(y - y_1 = m(x - x_1)\), plug in \(m = \frac{1}{2}\), \(x_1 = -2\), and \(y_1 = 5\). This gives: \[y - 5 = \frac{1}{2}(x + 2)\].

Step 4: Convert to Slope-Intercept Form

Distribute \(\frac{1}{2}\) on the right-hand side: \[y - 5 = \frac{1}{2}x + 1\]. Add 5 to both sides to solve for \(y\) in terms of \(x\): \[y = \frac{1}{2}x + 6\].

Step 5: Write the Final Equation

The slope-intercept form of the line is \(y = \frac{1}{2}x + 6\). This line has a slope of \(\frac{1}{2}\) and it passes through the point \((-2, 5)\).

Key Concepts

Linear Equations

Linear equations are a fundamental part of algebra, describing a straight line when graphed on a coordinate plane. At the core, these equations take the form \(y = mx + c\), where:

• \(m\) represents the slope of the line.
• \(c\) is the y-intercept, the point where the line crosses the y-axis.

To understand linear equations deeply, remember that they show a constant rate of change. This means if you increase or decrease \(x\) by a certain amount, \(y\) will change by a consistent amount proportional to the slope. This predictable relation is why linear equations are widely used in different fields whenever a straight-line relation is required, such as in physics for velocity or economics for cost analysis. Each linear equation represents a specific line on the coordinate plane.

Slope-Intercept Form

The slope-intercept form is a convenient way to write the equation of a line. It is expressed as \(y = mx + c\), where understanding its components can greatly aid in graphing lines quickly.

• Slope (\(m\)): Indicates the line's steepness and direction. A positive slope means the line rises from left to right, while a negative slope means it falls.
• Intercept (\(c\)): The y-coordinate at which the line crosses the y-axis.

This form is particularly useful because it directly reveals the slope and intercept, making it easy to graph. You simply start at the intercept on the y-axis and use the slope to find another point by moving up or down and left or right along the graph. This method provides a quick graphical representation of the linear relationship described by the equation.

Perpendicular Lines

Perpendicular lines are two lines that intersect at a right angle, forming a 90-degree angle between them. In the context of linear equations, understanding the relationship between their slopes is key.

• The slopes of two perpendicular lines are negative reciprocals of each other. If one line has a slope \(m\), the other will have a slope of \(-\frac{1}{m}\).

For example, if one line has a slope of \(-2\), a line perpendicular to it will have a slope of \(\frac{1}{2}\). Knowing this helps when constructing or analyzing perpendicular lines on a coordinate grid, as it is critical for applications such as navigation systems and architectural design. Finding perpendicular slopes ensures that the lines will geometrically meet at perfect right angles, an important concept not just in mathematics, but also in fields that require precision.

Point-Slope Form

The point-slope form of a linear equation is an alternative to the slope-intercept form and is especially useful when you know a point on the line and the slope. The general form is \(y - y_1 = m(x - x_1)\), where:

• \((x_1, y_1)\) are the coordinates of a specific point on the line.
• \(m\) is the slope of the line.

This form focuses on using a given point and slope to construct the equation, which is practical for deriving linear equations directly from data points. By substituting the known values of a point and the slope into the equation, you can swiftly rewrite it into slope-intercept form by solving for \(y\). This is a vital tool in algebra when dealing with problems involving multiple points or determining lines given specific points and slopes, whether in theoretical or real-world scenarios.