Problem 39 - Elementary Algebra Within Reach

Question

In Exercises 39-42, find \(m\) and \(b\) such that \(y=m x+b\) is the equation of the line through the points. $$ (2,-1),(6,1) $$

Step-by-Step Solution

Answer

The equation of the line passing through the given points (2, -1) and (6, 1) is: \(y = \frac{1}{2}x - 2\)

Step 1: Find the slope of the line (m)

The formula to find the slope (m) is: \[m = \frac{{y_2 - y_1}}{{x_2 - x_1}}\]So, substituting the given points into the formula, we get:\[m = \frac{{1 - (-1)}}{{6 - 2}} = \frac{1}{2}\]

Step 2: Find the y-intercept (b)

After finding the slope, the next step is to find the y-intercept. The 'y' intercept (b) can be found by substituting the value of m and one of the points into the equation \(y = mx + b\) and solving for 'b'. Let's choose the point (2, -1) for this calculation:\[-1 = \frac{1}{2} \cdot 2 + b \b = -1 - 1 = -2\]

Step 3: Write the equation of the line

Finally, substitute the values of 'm' and 'b' back into the equation \(y = mx + b\). The equation of the line passing through the points (2, -1) and (6, 1) is: \(y = \frac{1}{2}x - 2\)

Key Concepts

Slope-Intercept Form

Understanding the slope-intercept form of a linear equation is essential in algebra. This form is written as \( y = mx + b \), where \( m \) represents the slope, and \( b \) is the y-intercept of the line. The beauty of this form lies in its simplicity; it allows you to easily visualize the line by knowing just two pieces of information: how steep the line is (slope) and where it crosses the y-axis (y-intercept).

For example, if you have an equation like \( y = 3x + 2 \), you can immediately identify that the slope of the line is 3, meaning for each unit increase in \( x \), \( y \) increases by 3 units. The y-intercept is 2, indicating that the line crosses the y-axis at the point (0, 2). This form is incredibly helpful for graphing linear equations and analyzing linear relationships quickly and effectively.

Finding Slope

The slope of a line measures its steepness and direction. In other words, it tells us how much \( y \) changes for a change in \( x \). The slope is often denoted as \( m \) and can be found using the formula \( m = \frac{{\text{{rise}}}}{{\text{{run}}} = \frac{{y_2 - y_1}}{{x_2 - x_1}} \), where \( (x_1, y_1) \) and \( (x_2, y_2) \) are two distinct points on the line.

The process is straightforward. Subtract the y-coordinate of the first point from the y-coordinate of the second point, and do the same for the x-coordinates. Then, divide these two numbers to get the slope. For the line through the points \( (2,-1) \) and \( (6,1) \), the slope is calculated as \( m = \frac{1 - (-1)}{6 - 2} = \frac{2}{4} = \frac{1}{2} \). This fraction tells us that for every 2 units we move to the right along the x-axis, we move up 1 unit along the y-axis.

Y-Intercept

The y-intercept of a line is the point where the line crosses the y-axis. It's represented by the letter \( b \) in the slope-intercept equation \( y = mx + b \). Finding this value is a critical step in constructing a line's equation because it anchors the line at a specific location on the graph.

To determine the y-intercept, you can use one of the points on the line and the slope you've already found. Plugging these into the slope-intercept form and solving for \( b \) will yield the y-intercept. Let's take the point \( (2, -1) \) and a slope of \( \frac{1}{2} \) for instance. Plug these into the equation to get \( -1 = \frac{1}{2} \cdot 2 + b \), and solving for \( b \) gives us \( b = -1 - 1 = -2 \). This result tells us that our line's y-intercept is at the point \( (0, -2) \). Therefore, no matter where the line is or how it's slanted, it will always pass through this point on the y-axis.