Problem 6 Write an equation of the line in... [FREE SOLUTION]

Chapter 5: Problem 6

Write an equation of the line in point-slope form that passes through the given point and has the given slope. $$ (2,-1), m=3 $$

Short Answer

Expert verified

The equation of the line in point-slope form that passes through the point (2, -1) with slope 3 is $ y + 1 = 3x - 6 $.

Step by step solution

Analyze and arrange data

The point given is (2,-1) so x_1 = 2 and y_1 = -1. The slope given is m = 3.

Substitute the values into the equation

We substitute the given values into the point-slope equation y - y_1 = m(x - x_1). This will give us y - (-1) = 3(x - 2).

Simplify the equation

Simplify the equation to get the final form: y + 1 = 3x - 6.

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Key Concepts

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

Equation of a Line

Understanding how to write the equation of a line is an essential skill in algebra. A line can be represented by several different forms of equations, one of which is the point-slope form. When we talk about the equation of a line, we are establishing a mathematical way to describe every point that this line passes through. Lines can be described in three main forms:

Point-Slope Form: This form uses a known point on the line and the slope. By inserting these values into the equation $ y - y_1 = m(x - x_1) $, where $(x_1, y_1) $ are the coordinates of the point and $m$ is the slope, we can find the equation of the line.
Slope-Intercept Form: This form expresses lines in terms of their slope $m$ and y-intercept $b$. It's written as $y = mx + b $.
Standard Form: Another common way to express a line, written as $Ax + By = C$, where $A, B, $ and $C$ are integers.

Notably, knowing how to convert between these forms is useful for solving different types of problems.

Slope-Intercept Form

The slope-intercept form of a line's equation is particularly useful. It's written as $ y = mx + b $. Here, $m$ represents the slope of the line, showing how steep the line is, and $b$ represents the y-intercept, which is the point where the line crosses the y-axis. This form makes it easy:

To graph the equation: Start at the y-intercept $b$, and use the slope $m$, which indicates rise over run, to plot more points.
To understand the behavior of the line quickly: A positive $m$ means the line rises as you move from left to right, while a negative $m$ means it falls.

When using the slope-intercept form, conversion from other forms like point-slope is straightforward. For example, once you have the point-slope equation $y - y_1 = m(x - x_1) $, you can rearrange terms to get it into the y = mx + b form.

Linear Equations

Linear equations represent relationships between variables through straight lines on a graph. They are essential for expressing direct proportionality. A linear equation typically involves constants and variable terms that result in a straight line when plotted on a graph. These equations take forms like:

One-Variable Equations: Example: $2x = 6$ represents a vertical or horizontal line.
Two-Variable Equations: Example: $3x + 4y = 12$, when rearranged, shows a line in the coordinate plane.

Linear equations serve as foundational tools for modeling relationships where changes in one variable lead to changes in another at a consistent rate. They're pivotal in fields such as economics, physics, and basic algebraic studies.

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91影视

Write an equation of the line in point-slope form that passes through the given point and has the given slope. $$ (2,-1), m=3 $$

Short Answer

Step by step solution

Analyze and arrange data

Substitute the values into the equation

Simplify the equation

Key Concepts

Equation of a Line

Slope-Intercept Form

Linear Equations

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