Problem 47 - Algebra 1

Question

Write the point-slope form of the equation of the line that passes through the point and has the given slope. Then rewrite the equation in slope-intercept form. $$ (-9,-6), m=-\frac{2}{3} $$

Step-by-Step Solution

Answer

The point-slope form is: y + 6 = -2/3 * (x + 9). The slope-intercept form is: y = -2/3x -6.

Step 1: Insert values into the point-slope form

In the point-slope form, y - y1 = m(x - x1), replace m with -2/3, x1 with -9 and y1 with -6. That gives you the equation y - (-6) = -2/3 * (x- (-9)).

Step 2: Simplify the equation

Simplify the equation and you get y + 6 = -2/3 * (x + 9).

Step 3: Transform to the slope-intercept form

Transform the equation into the slope-intercept form y = mx + b, by isolating y. This gives you the equation y = -2/3x -6.

Key Concepts

Slope-Intercept Form

The slope-intercept form is a straightforward way to express the equation of a line. It is written as \( y = mx + b \). In this equation, \( m \) represents the slope of the line, while \( b \) stands for the y-intercept. The y-intercept is the point where the line crosses the y-axis. This form is particularly favored for graphing because it directly shows both the slope and the y-intercept.
To convert an equation into slope-intercept form, you isolate \( y \) on one side. This allows you to clearly identify the line's slope and y-intercept.
Benefits of using the slope-intercept form include:

• Easy identification of the slope and y-intercept.
• Simplified graph plotting.
• Quick comparison of different linear equations.

When dealing with linear equations, getting them into slope-intercept form can make subsequent mathematical operations more intuitive.

Linear Equations

Linear equations are equations that graph as straight lines on a Cartesian plane. These equations typically have the form \( y = mx + b \) in two variables, \( x \) and \( y \). A linear equation can be transformed into various forms including the standard form \( Ax + By = C \) and point-slope form \( y - y_1 = m(x - x_1) \).
The most distinctive feature of a linear equation is its constant rate of change, represented by the slope. This constant slope means the equation won't graph as curves but as straight lines with equal rise over run.
Some key characteristics of linear equations include:

• Constant slope (rate of change).
• Straight-line graph representation.
• Can be expressed in multiple forms for different mathematical needs.

Understanding linear equations is crucial as they are fundamental in algebra and pave the way for more complex concepts in mathematics.

Slope of a Line

The slope of a line is a measure of its steepness. It is usually denoted by \( m \) and calculated as the ratio of the rise over the run between two points on a line. Mathematically, it is expressed as \( m = \frac{y_2 - y_1}{x_2 - x_1} \).
The slope indicates how much \( y \) changes for every unit of change in \( x \). If the slope is positive, the line inclines upward, while a negative slope indicates a downward incline.
Some characteristics of slope include:

• Positive, negative, zero, or undefined slope types.
• Helps determine the direction and angle of a line.
• Essential in writing equations in point-slope and slope-intercept forms.

A zero slope means the line is perfectly horizontal, while an undefined slope is vertical. Comprehending slope is vital as it provides insights into how two different variables relate within a linear equation.