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Chapter 3: Problem 65

What is the point-slope form of the equation of a line?

Short Answer

Expert verified
The point-slope form is \( y - y_1 = m(x - x_1) \), where \((x_1, y_1)\) is a point on the line and \(m\) is the slope.

Step by step solution

01

Introduction to Point-Slope Form

The point-slope form of a line's equation is used when you know a point on the line and the slope of the line. This form is particularly useful because it allows for easy conversion to other forms, such as the slope-intercept form or the standard form.
02

Understanding the Formula

The point-slope form of the equation of a line is given as \( y - y_1 = m(x - x_1) \). In this equation, \((x_1, y_1)\) is a specific point on the line, and \(m\) represents the slope of the line.
03

Identify the Components

To write the equation of a line in point-slope form, identify a specific point on the line, \((x_1, y_1)\), and the slope of the line, \(m\). These will be used as the parameters in the equation.
04

Write the Equation

Insert the known values of \((x_1, y_1)\) and \(m\) into the formula \( y - y_1 = m(x - x_1) \). This gives you the equation of the line in point-slope form.

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Key Concepts

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

Equation of a Line
An equation of a line represents a straight path on a coordinate plane and can be described in various forms. The point-slope form, slope-intercept form, and standard form are all methods to express the equation of a line. Each form has its unique features and uses.
  • **Point-slope form** is useful when you know a point on the line and the line's slope.
  • **Slope-intercept form** makes it easy to identify the slope and the y-intercept.
  • **Standard form** organizes the equation into a neat, generalized arrangement.
Understanding each form helps in solving for a line's characteristics or converting between forms to suit different problem-solving contexts. This flexibility is particularly beneficial when addressing diverse mathematical problems including systems of equations and graphing tasks.
Slope-Intercept Form
The slope-intercept form of the equation of a line is widely used due to its simplicity and ease of interpretation. It is expressed as:\[y = mx + b\]Here, **\(m\)** denotes the slope of the line, and **\(b\)** is the y-intercept, which is where the line crosses the y-axis. This form is straightforward as it directly showcases the slope and y-intercept.
  • **Slope** \((m)\): Describes how steep the line is. A larger value indicates a steeper incline.
  • **Y-intercept** \((b)\): The point where the line crosses the y-axis. It provides a starting point for graphing the line.
Using the slope-intercept form, one can quickly graph a line by plotting the y-intercept and using the slope to determine the rise over run from that point. This form is immensely helpful in visual analyses and is often the go-to format for linear equations in algebra courses.
Standard Form
The standard form of a linear equation presents the equation as a clean and orderly statement:\[Ax + By = C\]In this formula, \(A\), \(B\), and \(C\) are integers, with \(A\) and \(B\) not both zero. The standard form is particularly useful in several mathematical scenarios such as calculating intersections.
  • **Uniformity**: Equations in standard form are often preferred for certain algebraic techniques, like the elimination method in systems of equations.
  • **Clarity**: By setting the equation to equal \(C\), you clearly define outcomes and constraints, useful in array and matrix calculations.
  • **Adaptability**: Easily converts to other forms such as point-slope or slope-intercept, depending on the problem's needs.
While it may seem more complex initially, mastering standard form enables you to tackle a variety of mathematical challenges with precision and ease.

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Most popular questions from this chapter

If the system represented by the augmented matrix at the right has no solution, what do you know about \(k ?\) Explain your answer. $$ \left[\begin{array}{llll} 1 & 1 & 0 & 1 \\ 0 & 0 & 1 & 2 \\ 0 & 0 & 0 & k \end{array}\right] $$

Use Cramer's rule to solve each system of equations. If a system is inconsistent or if the equations are dependent, so indicate. $$ \left\\{\begin{array}{l} x+y=1 \\ \frac{1}{2} y+z=\frac{5}{2} \\ x-z=-3 \end{array}\right. $$

Explain the difference between a matrix and a determinant. Give an example of each.

Determine whether each equation defines \(y\) to be a function of \(x .\) If it does not, find two ordered pairs where more than one value of \(y\) corresponds to a single value of \(x .\) $$ x=|y| $$

Write a system of three independent linear equations in two variables that has the solution \((-5,2)\)
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