Problem 1 - Introductory Algebra for College Students

Question

Find the slope and the \(y\) -intercept of the line with the given equation. $$y=3 x+2$$

Step-by-Step Solution

Answer

The slope of the line is 3 and the y-intercept is 2.

Step 1: Identify the Slope

The slope (m) is the coefficient of x in the equation. Looking at the equation \(y=3x+2\), the coefficient of x is 3. Therefore, the slope of the line is 3.

Step 2: Identify the Y-Intercept

The y-intercept (b) is the constant in the equation. In the given equation \(y=3x+2\), the constant is 2. So, the y-intercept of the line is 2.

Key Concepts

Linear Equation

A linear equation is a mathematical expression that forms a straight line when graphed on a coordinate plane. The basic form of a linear equation in two variables is known as the slope-intercept form, given by \(y = mx + b\). This form makes it easy to identify the key characteristics of the line, namely the slope and the y-intercept. In this structure:

• \(y\) represents the dependent variable, or the output of the line.
• \(x\) is the independent variable, the input you choose.
• \(m\) stands for the slope, which shows the rate of change between the variables.
• \(b\) reflects the y-intercept, the point where the line crosses the y-axis.

Linear equations are important because they allow us to model and solve real-world problems involving constant rates of change. They also provide a straightforward way to predict values and understand relationships between two quantities.

Slope

The slope of a line, denoted as \(m\), plays a crucial role in determining the direction and steepness of the line. In the slope-intercept form \(y = mx + b\), you'll spot the slope as the coefficient of \(x\). This number tells us how much \(y\) changes for each unit increase in \(x\).

• If the slope is positive, meaning \(m > 0\), the line rises as you move from left to right.
• A negative slope, where \(m < 0\), indicates the line falls as you progress to the right.
• When the slope is zero, the line remains flat, indicating no change in \(y\) no matter how \(x\) varies.

Understanding the slope allows you to quickly grasp important aspects of the line's behavior—it shows whether the line is increasing or decreasing and at what rate. In our given equation \(y = 3x + 2\), the slope is \(3\), showing that for every additional unit of \(x\), \(y\) increases by 3 units.

Y-Intercept

The y-intercept, represented by \(b\) in the slope-intercept form \(y = mx + b\), is the point where the line crosses the y-axis. This value is particularly useful because it tells us how high or low the line starts when \(x\) is zero. It provides insight into the initial conditions or starting point of the relationship described by the equation.

• The y-intercept is simply the constant term in the equation.
• You can find it by setting \(x = 0\) in the equation.
• It shows the value of \(y\) when the input \(x\) is equal to zero, which simplifies finding the starting point of the line.

In the equation \(y = 3x + 2\), the y-intercept is \(2\). This means when \(x = 0\), the value of \(y\) is \(2\). This tells us that the line crosses the y-axis at the point \((0, 2)\). Having a clear understanding of the y-intercept helps in graphing and interpreting the line's equation.