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Chapter 2: Problem 13

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

Short Answer

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

Step by step solution

01

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.
02

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 \( b = -6 \).
03

Substitute Into Slope-Intercept Form

Substitute the slope \( m = 3 \) and the y-intercept \( b = -6 \) into the slope-intercept form equation to get: \( y = 3x - 6 \).
04

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.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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.

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