Problem 5 - Introductory Algebra for College Students

Question

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 is \(5\).

Step 1: Identify the Slope

To start, identify the slope from the equation. In the equation \(y=-\frac{1}{2}x + 5\), the coefficient of \(x\) is the slope. Here, the slope \(m\) is \(-\frac{1}{2}\).

Step 2: Identify the y-intercept

Next, identify the y-intercept from the equation. In the equation \(y=-\frac{1}{2}x + 5\), the constant term is the y-intercept. Here, the y-intercept \(b\) is \(5\).

Key Concepts

Slope

In algebra, the slope of a line is a crucial concept that helps us understand how a line inclines or declines. You can think of it as the rate of change or the steepness of a line. When working with linear equations in the form \(y = mx + b\), the slope is represented by the coefficient \(m\) in front of the \(x\) term.

For the equation \(y = -\frac{1}{2}x + 5\), the slope \(m\) is \(-\frac{1}{2}\). This particular slope tells us that for every unit increase in \(x\), \(y\) decreases by \(\frac{1}{2}\). In simple terms, the line falls downward as it moves from left to right.

Some key points to remember about slope:

• If the slope is positive, the line ascends from left to right.
• If the slope is negative, the line descends from left to right.
• If the slope is zero, the line is horizontal, indicating no incline.
• An undefined slope means the line is vertical.

Understanding slope is vital in algebra because it influences how graphs of equations look on a coordinate plane.

Y-intercept

The y-intercept is another foundational concept in linear equations. It describes where the line crosses the \(y\)-axis on a graph. In the equation \(y = mx + b\), the y-intercept is the constant term \(b\).

For the given equation \(y = -\frac{1}{2}x + 5\), the y-intercept is \(5\). This means that the line will intersect the \(y\)-axis at the point \((0, 5)\). In other words, when \(x\) is zero, \(y\) will be \(5\).

Here are some basic facts about the y-intercept:

• The y-intercept is always on the \(y\)-axis (where \(x = 0\)).
• It tells us the starting point of the line when graphing.
• Every non-vertical line will have exactly one y-intercept.

Knowing the y-intercept helps you place your line accurately on a graph and is an important aspect of graphing linear equations.

Algebra Basics

To solve problems involving linear equations, understanding the basics of algebra is essential. Linear equations are often expressed in the form \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. This is known as the slope-intercept form.

Algebra basics involve manipulating equations to find unknown values and understanding how they represent different lines on a graph. Here’s a breakdown of what you should know:

• Linear equations create straight lines on a graph.
• The slope shows the direction and steepness of the line.
• The y-intercept shows where the line crosses the y-axis.

By mastering these concepts, you can confidently handle tasks such as finding the slope and y-intercept, and graphically representing equations. Algebra is like a toolkit - practice and understanding of the basics will equip you to solve a wide variety of mathematical problems.