Problem 95 - Precalculus Essentials

Question

Explain how to derive the slope-intercept form of a line's equation, \(y=m x+b,\) from the point-slope form $$y-y_{1}=m\left(x-x_{1}\right).$$

Step-by-Step Solution

Answer

The transformation from the point-slope form to the slope-intercept form of a line's equation involves simplifying \(y - y1 = m(x - x1)\) to the slope-intercept form \(y = mx + b\), where \(-mx1 + y1\) equals to \(b\).

Step 1: Understand forms of equations

There are many ways to write the equation of a line but the most common forms are point-slope form \(y - y1 = m(x - x1)\) and the slope-intercept form \(y = mx + b\). Both forms present the same line in different looks.

Step 2: Express the equation in point-slope form

Start with the equation \(y - y1 = m(x - x1)\) which is the point-slope form of the equation. Here, \(m\) is the slope of the line, \((x1, y1)\) is a specific point on the line.

Step 3: Simplify the equation

Now, distribute the term \(m\) on the right side of the equation to get \(y - y1 = mx - mx1\).

Step 4: Further simplification

Now, add \(y1\) to both sides to isolate \(y\) on one side. We get \(y = mx - mx1 +y1\).

Step 5: From point-slope to slope-intercept form

The equation \(y = mx - mx1 + y1\) taken from the point-slope form is actually in the slope-intercept form. Here, \(-mx1 + y1\) is equal to \(b\) (the y-intercept) in the slope-intercept form. So, it can also be written as \(y = mx + b\).

Key Concepts

Point-Slope Form

Understanding the point-slope form of a line's equation is crucial for anyone studying basic algebra and coordinate geometry. It is given by the equation \( y - y_1 = m(x - x_1) \), where \( m \) represents the slope of the line and \( (x_1, y_1) \) is a specific point through which the line passes.

It's a straightforward way of writing a line's equation when you have a point and the slope. If, for instance, a line passes through the point (3, 4) and has a slope of 2, the point-slope form of the equation would be \( y - 4 = 2(x - 3) \). This form is especially useful for quickly understanding the direction and steepness of the line, as well as finding another point on the line.

Equation of a Line

The equation of a line is a fundamental concept in algebra that represents a straight path on a coordinate plane. There are various ways to express it, with the most common being point-slope and slope-intercept forms. Each form provides unique insights into the properties of the line.

While the point-slope form focuses on the line's angle of inclination and a specific point it passes through, the slope-intercept form (\( y = mx + b \) where \( b \) is the y-intercept) gives direct information about the line's slope and where it crosses the y-axis. Equations of lines are essential in understanding how two variables relate and how changes in one can affect the other in a linear relationship.

Slope of a Line

The slope of a line is a measure of its steepness, generally represented by the letter \( m \). Mathematically, it's calculated as the ratio of the 'rise' over 'run' between two points on the line. If you take any two points on the line \( (x_1, y_1) \) and \( (x_2, y_2) \), the slope \( m \) can be found using the formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \).

A positive slope means the line is increasing, a negative slope means the line is decreasing, and a slope of zero means the line is horizontal. The magnitude of the slope determines the line's angle relative to the horizontal axis—the larger the magnitude, the steeper the line.

Y-intercept

The y-intercept of a line is the point where the line crosses the y-axis on a coordinate plane. It's an important concept because it gives a starting value in the absence of x, providing a clear visual representation of where a line originates on the graph.

In the slope-intercept form of a line (\( y = mx + b \)), the y-intercept is represented by \( b \). This form makes it particularly easy to identify the y-intercept as you simply look at the constant term. For example, if the equation of a line is \( y = 3x + 2 \), then the y-intercept is 2. This means that with an x-value of zero, the value of y would be 2, placing the intercept at the point (0, 2) on the graph.