Problem 8 - Algebra 1

Question

Write an equation of the line in slope-intercept form. The slope is \(4 ;\) the \(y\) -intercept is \(-6\)

Step-by-Step Solution

Answer

The equation of the line in slope-intercept form is \(y = 4x - 6\).

Step 1: Identify Slope and Y-intercept

By reading the problem, we identify that the slope \(m\) is \(4\) and the y-intercept \(c\) is \(-6\).

Step 2: Substitute the Slope and Y-intercept

Substitute the slope and y-intercept into the equation \(y = mx + c\). Here \(m = 4\) and \(c = -6\). So, the equation becomes \(y = 4x - 6\).

Key Concepts

Equation of a Line

When we talk about the **equation of a line**, one of the most common forms we use is called the "slope-intercept form." This format makes it straightforward to recognize both the slope of the line and where it crosses the y-axis. The general structure is given by \[y = mx + c\]where:

• y represents the value of the dependent variable.
• x denotes the independent variable.
• m is the slope of the line.
• c is the y-intercept of the equation.

This equation is widely used because it provides clear and immediate information about how the line behaves. Once you know the values of the slope and the y-intercept, you can easily graph the line or analyze its behavior across different points.

Slope

The **slope** of a line is a measure of how steep or flat a line is. It indicates the direction of the line and tells us how much the line rises or falls as we move from left to right. Think of it as the "tilt" of the line. The slope is expressed as a number, known as "m" in the slope-intercept form. If the slope is:

• Positive: The line rises as it moves from left to right. For example, a slope of 4 means that for every unit you move to the right on the x-axis, the line goes up by 4 units.
• Negative: The line falls as you move from left to right.
• Zero: The line is perfectly horizontal, having no upward or downward tilt.
• Undefined (infinite): In the case of vertical lines.

Understanding the slope helps in determining the angle and direction of a line, which are crucial in many mathematical problems and real-life applications such as predicting trends and understanding data patterns.

Y-Intercept

The **y-intercept** is a key feature of the slope-intercept equation, represented by the letter "c." It defines the point where the line crosses the y-axis of a graph.This is essential because the y-intercept provides a starting point for plotting the line on a graph. For example, if the y-intercept is \(-6\), it means that when \(x = 0\), the value of \(y\) is \(-6\). Here are some simple facts about y-intercept:

• It's often the starting point when plotting a line on a graph.
• In real-world situations, it can represent initial conditions or starting values when measuring changes over time.
• Having a negative y-intercept means the line crosses below the origin (0,0) on the y-axis.
• If the y-intercept is zero, the line passes through the origin.

Understanding the y-intercept allows you to quickly graph lines and interpret initial values, making it easier to visualize and solve mathematical problems.