Problem 20 - Introductory and Intermediate Algebra for College Students 4th

Question

In Exercises \(13-26,\) begin by solving the linear equation for \(y .\) This will put the equation in slope-intercept form. Then find the slope and the \(y\) -intercept of the line with this equation. $$3 y=-9 x$$

Step-by-Step Solution

Answer

The slope of the line is -3 and the y-intercept is 0.

Step 1: Isolate \(y\)

Rewrite the equation \(3y = -9x\) and isolate \(y\) by dividing each side of the equation by 3. This gives \(y = -3x\).

Step 2: Identify the slope

In the slope-intercept form \(y = mx + c\), \(m\) is the coefficient of \(x\). So in our equation \(y = -3x\), the slope \(m = -3\).

Step 3: Identify the \(y\)-intercept

In the form \(y = mx + c\), \(c\) is the constant that denotes the \(y\)-intercept (the value where the line crosses the \(y\)-axis). There is no constant term in our equation other than -3 which is a coefficient of \(x\), thus it implies that the constant \(c = 0\).

Key Concepts

Linear Equations

The foundation of algebra includes understanding linear equations, which are mathematical expressions that describe a straight line when graphed on a coordinate plane. A standard form of a linear equation is given by the general formula \( Ax + By = C \), where \( A \), \( B \) and \( C \) are real numbers and \( x \) and \( y \) are variables.

However, to analyze the behavior of lines in a simple and intuitive way, the slope-intercept form is often used, which is written as \( y = mx + b \). In this form, \( m \) represents the slope of the line, indicating its steepness or flatness, while \( b \) is the y-intercept, the point at which the line crosses the y-axis. In the exercise provided, transforming the equation \( 3y = -9x \) into the slope-intercept form makes it easier to pinpoint these characteristics and gives us a clearer picture of the line's behavior.

Slope of a Line

The slope of a line is a measure of its steepness, often represented as \( m \) in the equation of a line. Mathematically, it’s calculated as the 'rise over run', or the change in \( y \) values divided by the change in \( x \) values between two points on the line. When looking at the slope-intercept form \( y = mx + b \), the slope \( m \) is simply the coefficient of \( x \).

Understanding Slope

If the slope is positive, the line rises from left to right. Conversely, if it's negative, the line falls. A slope of zero means the line is horizontal, indicating no rise regardless of the run. In the given exercise, the equation \( y = -3x \) has a slope of \( m = -3 \) suggesting a declining line that moves downwards as one travels from left to right on the graph.

Y-Intercept

The y-intercept is the location where the line crosses the y-axis. In the slope-intercept form \( y = mx + b \), \( b \) denotes the y-intercept's value. It tells us the point at which the x-coordinate is zero.

Significance of the Y-Intercept

The y-intercept is crucial because it provides a starting point for the line on a graph. It is represented by the coordinates \( (0, b) \). For instance, if a line has an equation of \( y = 4x + 2 \), the y-intercept is 2, meaning the line will cross the y-axis at the point \( (0, 2) \). In the example given from the textbook exercise, since the equation \( y = -3x \) does not have a constant term added to \( -3x \) the y-intercept is 0, placing the crossing point at the origin \( (0, 0) \) of the graph.