Problem 23 - Elementary Algebra

Question

For the following problems, determine the slope and \(y\) -intercept of the lines. $$ y=3 x+4 $$

Step-by-Step Solution

Answer

Answer: The slope (m) is 3 and the y-intercept (b) is 4.

Step 1: Identify the format of the linear equation

The given linear equation is in the format of y = mx + b, where m is the slope and b is the y-intercept. In this case, the equation is: $$ y = 3x + 4 $$

Step 2: Determine the slope (m)

Observe that the coefficient of x in the equation is the slope (m). In this case, the slope is 3: $$ m = 3 $$

Step 3: Determine the y-intercept (b)

Observe that the constant term in the equation is the y-intercept (b). In this case, the y-intercept is 4: $$ b = 4 $$

Step 4: State the slope and y-intercept

The slope of the line is 3 and the y-intercept is 4. Therefore, for the given equation \(y = 3x + 4\), the slope (m) is 3 and the y-intercept (b) is 4.

Key Concepts

Understanding Slope

The slope of a line is a measure of its steepness. This is important because it tells us how much the line rises or falls as we move from left to right. The slope is represented by the letter \( m \) in the linear equation format \( y = mx + b \). In simple terms:

• If the slope is positive, the line will rise as it moves to the right.
• If the slope is negative, the line will fall as it moves to the right.
• If the slope is zero, the line is perfectly horizontal and doesn't rise or fall at all.

In our exercise, the slope of the equation \( y = 3x + 4 \) is 3. Since it is positive, the line rises as it moves from left to right. The number 3 indicates that for every unit we move to the right, the line goes up by 3 units. Understanding the slope helps in predicting how the line behaves within a graph.

Clarifying Y-Intercept

The y-intercept is the point where the line crosses the y-axis on the graph. This is a crucial concept because it tells us the value of \( y \) when \( x = 0 \). In the linear equation format \( y = mx + b \), the y-intercept is denoted by \( b \).
In the given equation \( y = 3x + 4 \), the y-intercept is 4. This means that when \( x = 0 \), \( y \) equals 4, and thus the line intersects the y-axis at that point.
Knowing the y-intercept allows us to understand where the line begins on a graph. It is the starting point before the line begins to rise or fall according to the slope.

Equation Format of Linear Equations

The standard format of a linear equation is \( y = mx + b \), which is straightforward yet powerful in representing a line. Here is how each component of this equation format is defined:

• \( y \) is the dependent variable, meaning its value depends on the value of \( x \).
• \( m \) represents the slope, showing the rate of change between \( y \) and \( x \).
• \( x \) is the independent variable, which can be altered or controlled.
• \( b \) is the y-intercept, the value at which the line crosses the y-axis.

Understanding this equation format becomes essential as it directly reveals the slope and y-intercept, making it easy to quickly assess a line's characteristics. For instance, in the equation \( y = 3x + 4 \), you can immediately find that the slope is 3 and the y-intercept is 4 by looking at the terms of the equation. This format helps simplify graphing as well as comparing different linear equations.