Problem 26 - Precalculus Essentials

Question

Use the given conditions to write an equation for each line in point-slope form and slope-intercept form. Passing through (3,5) and (8,15)

Step-by-Step Solution

Answer

The point-slope form for the line passing through points (3,5) and (8,15) is \(y - 5 = 2(x - 3)\). The slope-intercept form of the same line is \(y = 2x - 1\).

Step 1: Finding The Slope

Considering the two points (3,5) and (8,15), the first step is to find the slope of the line, 'm'. We use the slope formula: \(m = (y_2 - y_1) / (x_2 - x_1)\). Substituting the given points into this equation gives us \(m = (15 - 5) / (8 - 3) = 2\). So, our slope 'm' is 2.

Step 2: Finding The Point-Slope Form

Next, using the slope 'm' and one of the points, say (3,5), we can derive the equation in point-slope form: \(y - y_1 = m(x - x_1)\). This, with our known values, becomes \(y - 5 = 2(x - 3)\). This is the point-slope form of the line.

Step 3: Finding The Slope-Intercept Form

Finally, we can transform the point-slope form into the slope-intercept form \(y = mx + b\). Doing this gives us \(y = 2x - 1\). So, the slope-intercept form of the line is \(y = 2x - 1\).

Key Concepts

Point-Slope Form

Point-slope form is a way to write the equation of a line when we know a point on the line and its slope. The formula is:

• \( y - y_1 = m(x - x_1) \)

Here, \((x_1, y_1)\) is a known point and \(m\) is the slope. This form is particularly useful when we have one point the line passes through and we know the slope. For example, if a line passes through the point \((3, 5)\) and has a slope of \(2\), we plug these values into the formula to get:

• \( y - 5 = 2(x - 3) \)

This equation tells us that as \(x\) changes, \(y\) changes in conjunction with a slope of \(2\), keeping in mind the starting point \((3, 5)\). The result is the point-slope form of the line, which is particularly handy to use directly when you first find the slope.

Slope-Intercept Form

The slope-intercept form of a line is one of the most popular ways to express a line equation. It clearly shows two significant parts: the slope and the y-intercept, in the equation:

• \( y = mx + b \)

Here, \(m\) represents the slope and \(b\) is the y-intercept, which is where the line crosses the y-axis. Starting from the point-slope form \( y - 5 = 2(x - 3) \), we can easily rearrange it to get the slope-intercept form. By solving for \(y\), we find:

• First, distribute the slope: \(y - 5 = 2x - 6\)
• Then, add \(5\) to both sides: \(y = 2x - 1\)

Here, \(2\) is the slope and \(-1\) is the y-intercept. This slope-intercept form is great for quickly graphing the line or comparing it with other lines, as it clearly outlines the line's angle and where it will cross the y-axis.

Finding the Slope

The slope of a line is a measure of its steepness and direction. It is calculated by finding the ratio of the change in \(y\)-coordinates to the change in \(x\)-coordinates between two points. The slope formula is:

• \( m = \frac{y_2 - y_1}{x_2 - x_1} \)

Using this formula with the points \((3, 5)\) and \((8, 15)\), we calculate:

• Change in \(y\): \(15 - 5 = 10\)
• Change in \(x\): \(8 - 3 = 5\)
• Slope \(m\): \(\frac{10}{5} = 2\)

Thus, the slope \(m\) is \(2\), meaning for every 1 unit increase in \(x\), \(y\) increases by 2 units. Understanding the slope is vital because it tells us how steep the line is and in what direction it moves. If the slope is positive, as in our case, the line rises as it moves from left to right. Conversely, a negative slope would mean the line falls as it moves across.