Problem 27 - Introductory and Intermediate Algebra for College Students 4th

Question

Write the point-slope form of the equation of the line satisfying each of the conditions in Exercises \(1-28 .\) Then use the point-slope form of the equation to write the slope-intercept form of the equation. \(x\) -intercept \(=-\frac{1}{2}\) and \(y\) -intercept \(=4\)

Step-by-Step Solution

Answer

The slope of the line is 8. The equation of the line in point-slope form is \(y = 8x + 4\), which is also the line's equation in slope-intercept form.

Step 1: Determine the Slope

The slope-intercept form of a line is given by the formula \(y = mx + b\), where \(m\) is the slope of the line and \(b\) is the y-intercept. For two points \((x1, y1)\) and \((x2, y2)\), the slope \(m\) can be calculated using the following formula: \(m = (y2 - y1) / (x2 - x1)\). So, for the points given as x-intercept \((-1/2,0)\) and y-intercept \((0,4)\), we have: \(m = (4-0) / (0 - (-1/2)) = 4 / 0.5 = 8\)

Step 2: Write the Equation in Point-Slope Form

A line equation in point-slope form is given by the formula \(y-y1 = m(x-x1)\), where \((x1, y1)\) is a point on the line. For the slope \(m=8\) and point \((-1/2,0)\), the equation becomes: \(y-0 = 8*(x-(-1/2)) => y = 8x + 4\)

Step 3: Convert to Slope-Intercept Form

The point-slope form equation \(y = 8x + 4\) is already in the slope-intercept form \(y = mx + b\), as we can see that the slope \(m\) is 8 and the y-intercept \(b\) is 4

Key Concepts

Slope-Intercept Form

The slope-intercept form is a way of writing linear equations, which is very user-friendly in graphing a line. It is given by the formula \(y = mx + b\). Here, \(m\) represents the slope of the line, which indicates the steepness or incline of the line. The value \(b\) is the y-intercept, which tells us where the line crosses the y-axis. These characteristics make it easy to draw a line on a graph once you have the slope and y-intercept.

For instance, if you have an equation in slope-intercept form like \(y = 2x + 3\), it means the slope \(m\) is 2. This means for every 1 unit increase in \(x\), \(y\) increases by 2 units. The line crosses the y-axis at \(b = 3\).

Overall, this form is favored for its simplicity and ability to quickly indicate both the slope and the y-intercept, helping to visualize the line in the coordinate plane with ease.

Line Equation

A line equation is a mathematical expression that describes a straight line on a graph. The equation lays out the relationship between the \(x\) and \(y\) coordinates of each point on the line. There are various forms of line equations, with the point-slope form and slope-intercept form being the most common.

• Point-Slope Form: \(y - y_1 = m(x - x_1)\) uses a known point on the line \((x_1, y_1)\) and the slope \(m\). It is particularly useful when you have a point and a slope and need to find the equation of the line.
• Slope-Intercept Form: Once you manipulate the point-slope form, you often aim to transform it into \(y = mx + b\), for ease of graphing and analysis.

Understanding line equations is essential for solving problems related to linear relationships and can be used in various applications such as predicting trends or analyzing data.

Intercepts

Intercepts are key elements of a line equation as they define where the line crosses the axes of a graph. There are two types of intercepts:

• x-intercept: This is the point where the line crosses the x-axis. At this point, the \(y\)-coordinate is zero. To find it, you'd generally set \(y = 0\) in the equation and solve for \(x\).
• y-intercept: This is the point where the line crosses the y-axis. At this point, the \(x\)-coordinate is zero. You find this intercept by setting \(x = 0\) in the equation and solving for \(y\).

Intercepts provide a quick and easy way to graph a line since they give you two key points of intersection with the axes that can guide your graph drawing. For example, from our original problem, we had \(x\)-intercept at \(-\frac{1}{2}\) and \(y\)-intercept at 4, allowing us to quickly graph the line in a coordinate plane.