Problem 17 - Algebra 1: Concepts and Skills

Question

Find the slope and y-intercept of the graph of the equation. $$y=6 x+4$$

Step-by-Step Solution

Answer

The slope of the graph is 6 and the y-intercept is 4.

Step 1: Identifying the Slope

The slope (m) is the coefficient of x in the given equation. In other words, it is the number that multiplies x. In the equation \( y = 6x + 4 \), the coefficient of x is 6, so the slope (m) is 6.

Step 2: Identifying the Y-Intercept

The y-intercept (b) is the constant in the given equation. In the equation \( y = 6x + 4 \), the constant is 4, so the y-intercept (b) is 4.

Key Concepts

Understanding Slope

The slope of a line is a measure of its steepness and is represented by the letter \( m \) in the slope-intercept form of a linear equation. Essentially, it tells us how much the \( y \)-value, or output, of a graph changes for each change in the \( x \)-value, or input.

The formula for the slope between two points \((x_1, y_1)\) and \((x_2, y_2)\) on a line is:

• \( m = \frac{y_2 - y_1}{x_2 - x_1} \)

This equation tells us the vertical change (rise) over the horizontal change (run) between two points.

In a graph, a positive slope means the line is rising as it moves from left to right, while a negative slope means the line is falling. Understanding slope is crucial for discerning how changes in one variable affect another in a linear relationship.

Identifying the Y-Intercept

The y-intercept of a line is the point at which the line crosses the y-axis. It is represented by the letter \( b \) in the slope-intercept form of a linear equation. This element of a graph is crucial as it tells us the value of \( y \) when \( x \) is zero.

In simpler terms, at the y-intercept, you can imagine where the line would "hit" the vertical y-axis if extended or drawn correctly.

• The y-intercept is given in the equation \( y = mx + b \) as the constant term \( b \).

For example, in the equation \( y = 6x + 4 \), the number 4 is the y-intercept. This means when \( x = 0 \), \( y \) will be exactly 4. Understanding the y-intercept helps to easily plot and interpret linear equations.

Exploring Linear Equations

Linear equations form straight lines on a graph and are among the most fundamental concepts in algebra and mathematics. They express a consistent relationship between two variables, typically \( x \) and \( y \). The general form of a linear equation is known as the slope-intercept form and is given by:

• \( y = mx + b \)

This format directly exhibits two important features of the line:

• The slope \( m \)
• The y-intercept \( b \)

Linear equations are widely used to model real-world phenomena where a constant rate of change is observed. Examples include calculating speed (as distance over time), projecting financial trends, and converting units.

By understanding linear equations, you can predict and solve problems involving direct relationships between quantities.