Problem 35 - Elementary Algebra

Question

Determine the slope and \(y\) -intercept of the lines. $$ y=-6 x $$

Step-by-Step Solution

Answer

Answer: The slope of the line is -6, and the y-intercept is 0.

Step 1: Identify The Slope-Intercept Form Of The Line Equation

The slope-intercept form of the line is \(y = mx + b\). Our line equation is given by: $$y = -6x$$.

Step 2: Identify The Slope m And Y-Intercept b In The Line Equation

In \(y=-6x\), there is no added constant. We can write it in the slope-intercept form with \(m=-6\) and \(b=0\). Therefore, our equation will be \(y = -6x + 0\).

Step 3: State The Slope And Y-Intercept

From the equation \(y = -6x + 0\), we have determined that the slope (m) is -6 and the y-intercept (b) is 0. So the slope of the line is -6, and the y-intercept is 0, which corresponds to the point (0, 0) on the graph.

Key Concepts

Slope

The slope in a linear equation is a crucial element that expresses the steepness or incline of a line. When we talk about the slope, represented by the letter \(m\), we are discussing how much the line rises or falls as it moves from left to right on a graph. It is detailed using the slope formula: \(m = \frac{\Delta y}{\Delta x}\). Here, \(\Delta y\) represents the change in the \(y\)-values, and \(\Delta x\) represents the change in the \(x\)-values.
In the context of the given equation \(y = -6x\), the slope \(m\) is \(-6\). This negative value indicates that the line slopes downward as it moves from the left to the right. For each unit increase in \(x\), the value of \(y\) decreases by 6 units. This tells us about the direction and the steepness of the line:

• A negative slope means the line goes down as it moves to the right.
• A larger absolute value of the slope (like \(-6\) versus \(-2\)) means a steeper line.

Understanding the slope helps in predicting how changes in one variable can affect the other in a linear relationship.

Y-Intercept

The \(y\)-intercept \(b\) of a linear equation in slope-intercept form \(y = mx + b\) is the point where the line crosses the \(y\)-axis. This occurs when \(x = 0\), making the intercept essentially \(b\) in the equation.
In our specific equation, \(y = -6x\), the \(y\)-intercept \(b\) is \(0\). Therefore, the line crosses the \(y\)-axis at the origin, which is marked as the point \((0, 0)\) on a graph. This particular position suggests that at the start, when there is no change in \(x\), our value for \(y\) is 0.
Knowing the \(y\)-intercept allows us to graph a line quickly and understand its initial point on the coordinate plane:

• The \(y\)-intercept is where the line begins, offering a starting point for graphing.
• An intercept of 0 implies the line goes directly through the origin as seen in this case.

Grasping the concept of \(y\)-intercept provides insight into how a line behaves on the graph.

Linear Equations

Linear equations form the foundation for graphing lines and understanding relationships between variables in algebra. They are typically written in the slope-intercept form: \(y = mx + b\). This structure makes it easier to analyze and sketch lines on a coordinate plane by readily identifying the slope and \(y\)-intercept.
The equation \(y = -6x\) is a perfect example of a linear equation. It demonstrates a direct relationship between \(x\) and \(y\), wherein the value of \(y\) changes continuously as \(x\) changes. Here, the linearity implies:

• The graph of the equation is a straight line.
• Both the slope and intercept provide essential information for plotting the line.

Linear equations help in drawing connections between variables and understanding how adjustments in one can impact the other.
For instance, alterations in the slope can change the line's angle, while shifts in the \(y\)-intercept can move the line up or down on the graph. Comprehending these equations aids in decoding the structural relationships within data and scenarios expressed mathematically.