Problem 6 - Algebra 1

Question

Write an equation of the line in slope-intercept form. The slope is \(-2 ;\) the \(y\) -intercept is \(0 .\)

Step-by-Step Solution

Answer

The equation of the line in slope-intercept form is \(y = -2x\).

Step 1: Identify the given values

The given values are the slope \(m = -2\) and the y-intercept \(c = 0\).

Step 2: Recall the slope-intercept form of a line

The slope-intercept form is given by \(y = mx + c\).

Step 3: Substitute the given values

By substituting the given values \(-2\) for \(m\) and \(0\) for \(c\) into the formula, we get the equation of the line as \(y = -2x + 0\).

Step 4: Simplify the equation

The '+ 0' part can be eliminated in the equation, as adding zero to any variable won't change its value, hence the final equation of the line is \(y = -2x\).

Key Concepts

Slope-Intercept Form

When you come across linear equations in your math studies, one of the most commonly used forms is the slope-intercept form. This form is especially helpful for quickly graphing or understanding any straight line. The slope-intercept form of a line is expressed as \( y = mx + c \). This equation is convenient because it gives you a direct method to identify two critical components of the line: the slope \( m \) and the y-intercept \( c \).

In the formula:

• \( y \) represents the dependent variable or the vertical coordinate.
• \( x \) stands for the independent variable or the horizontal coordinate.
• \( m \) denotes the slope of the line.
• \( c \) symbolizes the y-intercept, which is the point where the line intersects the y-axis.

This format is straightforward and is often your go-to choice whenever you need to describe a line in an easy-to-visualize manner.

Slope

The slope of a line is a crucial concept in understanding how linear equations work. It describes how steep the line is and in which direction it tilts. A line can either ascend, descend, or remain constant, which is all depicted by its slope.

Here are key points about slope:

• It is symbolized by \( m \) in the slope-intercept form.
• Numerically, it tells you the change in \( y \) when \( x \) changes by one unit. Essentially, it is the "rise over run".
• A positive slope means the line ascends or goes upwards as you move from left to right.
• A negative slope implies the line descends or goes downwards when you move from left to right.
• If the slope is zero, the line is horizontal, indicating no change in \( y \) at all.

Thus, in the equation \( y = -2x \), a slope of \(-2\) shows that for every unit increase in \( x \), \( y \) decreases by two units, leading the line to tilt downwards.

Y-Intercept

The y-intercept is another fundamental part of the slope-intercept equation. It provides a starting point for plotting the line on a graph because it tells where the line crosses the y-axis.

Understanding y-intercepts:

• It is represented by the constant \( c \) in the equation \( y = mx + c \).
• The y-intercept is the value of \( y \) when \( x \) is zero. This makes it relatively easy to find on a graph.
• In the given problem, the y-intercept is 0, denoting that the line goes through the origin of the graph \((0, 0)\).

This simple point helps to position the line correctly on a graph. By combining the y-intercept with the slope, you can effortlessly sketch the entire line or gain insights into the linear relationship.