Problem 5 - Introductory and Intermediate Algebra for College Students 4th

Question

In Exercises \(1-12,\) find the slope and the \(y\) -intercept of the line with the given equation. $$y=-\frac{1}{2} x+5$$

Step-by-Step Solution

Answer

The slope of the line is \(-\frac{1}{2}\) and the y-intercept of the line is \(5\).

Step 1: Identify the Slope

The slope of the line is the coefficient of \(x\) in the equation. In this case, the slope (\(m\)) is \(-\frac{1}{2}\).

Step 2: Identify the y-intercept

The y-intercept is where the line crosses the y-axis. This is the constant in the equation. In this case, the y-intercept (b) is \(5\).

Key Concepts

Slope

In the context of linear equations, the slope is a critical concept that describes the line's steepness and direction. Imagine you are hiking up a hill; the slope tells you how steep the hill is. In mathematical terms, the slope is also known as the "rate of change" since it shows how much the "y" value changes with a change in "x". This is why in the equation of a straight line, often described as the slope-intercept form, \( y = mx + b \), the slope is represented by the letter \(m\). Here, the equation \( y = - \frac{1}{2} x + 5 \) indicates that the slope is \(-\frac{1}{2}\). This means for every 1 unit increase in \(x\), \(y\) decreases by \(\frac{1}{2}\).

• The negative sign in the slope indicates the line is descending left to right.
• A zero slope means a horizontal line.
• A positive slope points to an upward rising line.

Understanding slope helps you predict how a variable changes in relation to another, which is not just crucial in math but also in fields like economics and science.

Y-intercept

The y-intercept is another key component of a linear equation. It highlights where the line crosses the y-axis. Imagine starting a race; the y-intercept is your starting point on the y-axis. In equations written in the slope-intercept form, such as \( y = mx + b \), the y-intercept is denoted by \(b\). In our example, \( y = - \frac{1}{2} x + 5 \), the y-intercept is \(5\). This means regardless of the value of \(x\), when \(x=0\), \(y\) is already at \(5\).

• If the y-intercept is positive, the line starts above the origin on the axis.
• If it's negative, the line starts below the origin.

This concept is especially useful when graphing, as it tells you precisely where to start plotting your line on a graph.

Coefficient of x

In a linear equation, the coefficient of \(x\) plays a vital role. It is the number placed in front of the \(x\) variable, and it directly represents the slope of the line. Thus, it has a dual function, both indicating how steep the line is and the direction it goes. In the equation \( y = -\frac{1}{2} x + 5 \), the coefficient of \(x\) is \(-\frac{1}{2}\), matching the slope of the line.

The coefficient helps you understand:

• The line's angle concerning the x-axis.
• Whether the line rises or falls as you move along the x-axis.
• How each unit change in \(x\) affects \(y\).

Recognizing the coefficient of \(x\) allows you to quickly determine the slope and thus better understand the characteristics of the line defined by the equation.