Problem 32 - Algebra 1: Concepts and Skills

Question

Write in point-slope form the equation of the line. Then rewrite the equation in slope-intercept form. $$ (6,2), m=\frac{1}{2} $$

Step-by-Step Solution

Answer

The equation of the line in slope-intercept form is \(y = \frac{1}{2}x - 1\).

Step 1: Write Equation in Point-Slope Form

Insert the given point (6,2) and slope m = 1/2 into the point-slope formula. The equation becomes: \(y - 2 = \frac{1}{2}(x - 6)\).

Step 2: Distribute the Slope

Multiply it out to get \(y - 2 = \frac{1}{2}x - 3\).

Step 3: Transform into Slope-Intercept Form

Add 2 on both sides to solve for y in terms of x. The equation becomes \(y = \frac{1}{2}x - 1\) which is the slope-intercept form.

Key Concepts

Understanding Point-Slope Form

Point-slope form is a straightforward and powerful way to express the equation of a line when you know one point on the line and the slope. This form is especially useful for writing the equation quickly without complicated calculations. The general formula for point-slope form is:

• \( y - y_1 = m(x - x_1) \)

Here, \((x_1, y_1)\) represents the coordinates of a point the line passes through, and \(m\) is the slope of the line. For example, if you know that a line passes through the point \((6, 2)\) and has a slope \( \frac{1}{2} \), you can plug these values into the formula to find the equation. This results in:

• \( y - 2 = \frac{1}{2}(x - 6) \)

This equation can be further manipulated to other forms, like the slope-intercept form, but its main advantage is its direct relation to specific line properties.

Transforming to Slope-Intercept Form

The slope-intercept form of a line equation is another valuable and popular format you should be familiar with. It allows you to see key details about the line at a glance. In this form, the general equation is:

• \( y = mx + b \)

Here, \(m\) is the slope, similar to in point-slope form, and \(b\) is the y-intercept, or the value of \(y\) when \(x = 0\). From our point-slope form equation \( y - 2 = \frac{1}{2}(x - 6) \), we can convert it to slope-intercept form by distributing the slope and solving for \(y\). First, expand the equation to get \( y - 2 = \frac{1}{2}x - 3 \). Then, add 2 to both sides to get \( y = \frac{1}{2}x - 1 \). Here, \( \frac{1}{2} \) is the slope, and \(-1\) is the y-intercept, making it easy to plot the line on a graph.

Delving into the Slope

The slope is a crucial concept in understanding linear equations and lines. It measures the steepness of the line and indicates the direction in which the line inclines or declines.

• If the slope \(m\) is positive, the line rises as it moves from left to right.
• If the slope \(m\) is negative, the line falls as it moves from left to right.
• A larger value of \(m\) indicates a steeper incline or decline.
• If \(m = 0\), the line is perfectly horizontal.

In our example, the slope \(m\) is \( \frac{1}{2} \). This means the line rises half a unit for each full unit it moves to the right. Understanding the slope helps in visualizing how the line traverses through the graph, providing insights into its rate of change. This is why the slope is often considered one of the most informative aspects of a linear equation.