Problem 3 - Introductory Algebra for College Students

Question

Write the point-slope form of the equation of the line satisfying each of the conditions in Exercises. Then use the point-slope form of the equation to write the slope-intercept form of the equation. Slope \(=5,\) passing through \((-2,6)\)

Step-by-Step Solution

Answer

The point-slope form of the given line is \(y - 6 = 5(x + 2)\) which simplifies to \(y - 6 = 5x + 10\). The slope-intercept form of the line is \(y = 5x + 16\).

Step 1: Writing in Point-slope form

Point-slope form of a line's equation is: \(y - y_{1} = m(x - x_{1})\), where \(m\) is the slope of the line and \((x_{1}, y_{1})\) are the coordinates of a given point that the line passes through. Substituting the given point \((-2,6)\) into the equation and the given slope \(5\), we have \(y - 6 = 5(x - (-2))\).

Step 2: Simplifying the Point-slope form

The equation from step 1: \(y - 6 = 5(x + 2)\), simplifies to \(y - 6 = 5x + 10\)

Step 3: Writing in Slope-intercept form

The slope-intercept form of a line's equation is: \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. Solving the equation from step 2 for \(y\) to obtain the slope-intercept form, we have \(y = 5x + 16\).

Key Concepts

Slope-Intercept Form

The slope-intercept form is a way of expressing the equation of a straight line. This way is great because it lets you immediately see the slope and the y-intercept. The formula is written as: \[ y = mx + b \] where:

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

Understanding this form can make it easier for you to graph a line quickly and recognize how it fits into your overall understanding of equations and graphs.
By looking at the form, if \( m \) is positive, the line will tilt upwards as it moves from left to right. If \( m \) is negative, it will slope downwards.
In relation to the exercise, converting from point-slope form to slope-intercept form helps to reveal the y-intercept (\( b \) in the formula), which is essential for graphing the line.

Linear Equations

Linear equations describe straight lines. They are a fundamental concept in algebra. Understanding them is crucial because they appear in various forms and are used to solve real-world problems.
A linear equation can often be recognized in forms such as:

• Slope-intercept form: \( y = mx + b \)
• Point-slope form: \( y - y_1 = m(x - x_1) \)
• Standard form: \( Ax + By = C \)

Each of these formats is valuable for different scenarios.
For instance, if you need to find the slope quickly, slope-intercept form works best.
If you have a specific point and slope, point-slope form comes in handy. Knowing how to convert between these forms is key. It provides flexibility in working with linear equations.

Slope of a Line

The slope of a line is a measure of its steepness and direction. It tells us how much \( y \) changes for a given change in \( x \).
Slope is usually represented by the letter \( m \) in equations. It can be calculated using the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] where \((x_1, y_1)\) and \((x_2, y_2)\) are two distinct points on the line.
The slope can tell us a lot about the line:

• A positive slope means the line rises as it moves from left to right.
• A negative slope means the line falls.
• A zero slope means the line is horizontal, showing no change in \( y \) as \( x \) changes.
• An undefined slope means the line is vertical, and \( x \) stays the same regardless of \( y \).

In the exercise, the slope is 5, indicating a line that rises steeply to the right. This plays a crucial part in both point-slope and slope-intercept forms, guiding how the line behaves.