Problem 13 - Algebra 2

Question

Write an equation in slope-intercept form for the line that satisfies each set of conditions. slope \(3,\) passes through \((0,-6)\)

Step-by-Step Solution

Answer

The equation is \( y = 3x - 6 \).

Step 1: Understand Slope-Intercept Form

The slope-intercept form of a line is given by the equation \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.

Step 2: Identify Given Information

We are given a slope \( m = 3 \) and a point on the line \( (0, -6) \). The point \( (0, -6) \) tells us that the line crosses the y-axis at \( -6 \), so \( b = -6 \).

Step 3: Substitute Into Slope-Intercept Form

Substitute the slope \( m = 3 \) and the y-intercept \( b = -6 \) into the slope-intercept form equation to get: \( y = 3x - 6 \).

Step 4: Verify the Equation

To verify, ensure that the point \( (0, -6) \) lies on the line. Substitute \( x = 0 \) into the equation \( y = 3x - 6 \) to get \( y = -6 \). The point satisfies the equation, confirming it is correct.

Key Concepts

Linear Equation

A linear equation represents a straight line when plotted on a graph. These equations are fundamental in algebra, and they come in various forms, such as standard form, point-slope form, and slope-intercept form. Among these, the slope-intercept form is widely used due to its simplicity and ease of visual interpretation.

The equation of a line in slope-intercept form is written as:

• \( y = mx + b \)

where:

• \( y \) is the dependent variable.
• \( x \) is the independent variable.
• \( m \) is the slope of the line.
• \( b \) is the y-intercept, the point where the line crosses the y-axis.

Understanding these components is key when writing or decoding a linear equation, as it provides a clear picture of the line's behavior and position on a coordinate plane.

Slope

The slope of a line is a measure of its steepness. It tells us how much the line rises or falls as we move from left to right across a graph. In mathematical terms, the slope \( m \) is found by the change in \( y \) divided by the change in \( x \), often noted as "rise over run."

Here are the characteristics of the slope:

• A positive slope means the line rises as it moves to the right.
• A negative slope indicates the line falls as it moves to the right.
• If the slope is zero, the line is horizontal.

For the given equation, the slope is 3. This indicates that for every single unit increase in \( x \), \( y \) increases by 3 units. This positive slope results in an upward trend, depicting a line that angles upwards on a graph from left to right.

Y-Intercept

The y-intercept of a line is the point where the line crosses the y-axis. In the slope-intercept form equation \( y = mx + b \), the y-intercept is represented by \( b \).

Important aspects of the y-intercept include:

• It is the value of \( y \) when \( x \) is zero.
• It provides a starting point for drawing the line on a graph.

In our solved equation, \( y = 3x - 6 \), the y-intercept is \(-6\). This means that the line crosses the y-axis at the point \( (0, -6) \). The y-intercept is critical because it allows us to plot the line accurately on a graph by providing a fixed point through which the line must pass.

Understanding the y-intercept helps in grasping how lines shift up or down in a graph, maintaining the pattern dictated by their slope.