Problem 2 - Algebra 1: Concepts and Skills

Question

Identify the slope of the line that has the equation \(y=-4 x+15\)

Step-by-Step Solution

Answer

The slope of the line is -4.

Step 1: Identify the form of the given equation

The equation given, \(y= -4x + 15\), is in slope-intercept form \(y = mx + c\), where \(m\) is the slope and \(c\) is the y-intercept.

Step 2: Identify the slope

Looking at the provided equation, the coefficient of \(x\) is -4. This means our slope (\(m\)) is -4.

Key Concepts

Slope-Intercept Form

Understanding the slope-intercept form of a linear equation is essential for graphing and analyzing lines efficiently. Specifically, this form is expressed as y = mx + b, where m represents the slope of the line and b indicates the y-intercept, the point at which the line crosses the y-axis.

Slope, given by m, measures the steepness of the line, showing the ratio of the vertical change (rise) to the horizontal change (run) between any two points on the line. For example, if the equation of a line is y = -4x + 15, the slope is -4. This shows that for each unit the line moves horizontally to the right, it moves 4 units down, since the slope is negative. The beauty of the slope-intercept form is in its simplicity, allowing you to quickly identify these key characteristics of a line with just a quick glance at the equation.

Linear Equations

Linear equations are foundational in algebra and represent straight lines in a two-dimensional space. The general form of a linear equation is ax + by = c, where a, b, and c are constants. These equations become immensely powerful when presented in the slope-intercept form, making it easy to graph and understand the behavior of the line.

For example, when we see the equation y = -4x + 15, it tells us immediately that the line is linear, which means it has a constant rate of change, and this change is represented by the slope, -4. Variations in the slope and y-intercept lead to different lines, positions, and angles relative to the axes. Understanding these variations can help students quickly sketch the graph of the equation without the need for plotting multiple points.

Y-Intercept

The y-intercept of a line is the point at which the line crosses the y-axis, which occurs when x = 0. In the slope-intercept form y = mx + b, it is represented by b. It's a crucial concept as it indicates the start of the line on the graph, essentially where the function 'begins' in the context of a two-dimensional Cartesian plane.

Returning to our example, the y-intercept for the equation y = -4x + 15 is 15. This signifies that the point (0, 15) lies on the line and is the location where the line will intersect with the y-axis. This piece of information combined with the slope provides a complete visual overview of the line's behavior in the coordinate system, as we understand both its angle (via the slope) and its starting point (via the y-intercept).