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Chapter 4: Problem 41

In Exercises \(41-44\), determine the slope of the line. $$ y-2=5(x+3) $$

Short Answer

Expert verified
The slope of the line is 5.

Step by step solution

01

Understand the point-slope form

The provided equation of the line is \(y - 2 = 5(x + 3)\). This is already in point-slope form, which is \(y - y_1 = m(x - x_1)\), where \(m\) is the slope and \((x_1, y_1)\) are the coordinates of any point on the line. In this equation, \(m\) is the coefficient of \(x\).
02

Identify the slope

Looking at the equation, notice that the number being multiplied by \(x\) is 5. Therefore, the slope \(m\) of the line is 5.

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Key Concepts

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

Point-Slope Form
The point-slope form is a way of expressing the equation of a line using its slope and a specific point on the line. It is represented by the equation: \( y - y_1 = m(x - x_1) \).This format clearly shows:
  • \( m \): the slope of the line, which tells us how steep the line is.
  • \( (x_1, y_1) \): the coordinates of a point on the line.
In practice, it helps to quickly write linear equations when you know one point and the slope.
This form is particularly helpful when solving problems involving linear equations in a straightforward and organized manner. It essentially describes how the "rise" (the change in \( y \)) over the "run" (the change in \( x \)) affects the line through a specific point.
Linear Equations
Linear equations are mathematical expressions that describe a straight line when graphed on a coordinate plane.
They are used widely in mathematics to model relationships between variables. A key characteristic of a linear equation is its constant slope, which tells us that the rate of change is consistent.
There are several common forms of linear equations:
  • Point-slope form: Useful for writing the equation of a line with a known slope and a point.
  • Slope-intercept form (\( y = mx + b \)): Useful for quickly identifying the slope \( m \) and the y-intercept \( b \).
  • Standard form (\( Ax + By = C \)): Used in many algebraic operations.
For the original exercise, the point-slope form is already given, making it simple to extract the slope and understand the line's characteristics.
Educational Exercises
Educational exercises are essential in reinforcing learning concepts by providing students with practical problems to solve.
They are designed to help students understand foundational skills such as identifying slopes, writing equations, and graphing lines.
Effective exercises should:
  • Present clear instructions to avoid misunderstandings.
  • Apply concepts to real-world situations to enhance engagement and relevance.
  • Include step-by-step solutions, helping students to follow the logic and reasoning involved in problem-solving.
By working through exercises like determining the slope from point-slope form, students solidify their grasp of linear equations and their applications in everyday scenarios.

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Most popular questions from this chapter

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In Exercises \(13-16\), sketch the graph of the linear inequality. $$ x \leq 3 $$

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