Problem 40 - Understanding Elementary Algebra with Geometry

Question

Determine the slope of the line from its equation. $$y=-4 x+2$$

Step-by-Step Solution

Answer

The slope is -4.

Step 1: Identify the Equation Form

Recognize that the given equation is in the slope-intercept form which is written as: \[y = mx + b\] where \(m\) represents the slope and \(b\) represents the y-intercept.

Step 2: Extract the Slope

In the given equation, \(y = -4x + 2\), compare it to the slope-intercept form equation \(y = mx + b\). The coefficient of \(x\) is \(-4\), so the slope \(m\) is \(-4\).

Key Concepts

slope-intercept form

When dealing with linear equations, the slope-intercept form is one of the most common and useful formats you'll encounter. The slope-intercept form is expressed as y = mx + b.

• y - The dependent variable (usually represents something you're trying to predict or measure).
• x - The independent variable (usually represents the input or cause).
• m - The slope of the line, which tells you how steep the line is and the rate at which y changes for a change in x.
• b - The y-intercept, where the line crosses the y-axis (vertical axis).

Understanding this form can make solving or graphing linear equations much simpler.

linear equations

Linear equations represent straight lines when plotted on a graph. These equations have the general form of y = mx + b.These lines are characterized by having a constant slope, meaning the rate of change between the two variables is steady.

• They can be written in different forms, such as slope-intercept form (y = mx + b) or standard form (Ax + By = C).
• They are fundamental in understanding relationships in algebra, as many real-world scenarios can be modeled with linear equations.

When you understand that each term in a linear equation plays a specific role, it becomes easier to interpret and manipulate the equation.

extracting the slope

Extracting the slope from an equation in the slope-intercept form is straightforward. Recall our generic slope-intercept form y = mx + b.To find the slope (m), you focus on the coefficient of x.

• Identify the term next to x. This coefficient represents the slope.
• In the equation y = -4x + 2, the term next to x is -4.
• Thus, the slope (m) in this equation is -4.

Always remember, the slope conveys how steep the line is and the direction it goes. A positive slope means the line rises, while a negative slope means it falls.