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Chapter 1: Problem 19

Find an equation of the line: with \(m=-5\), containing \((1,-5)\)

Short Answer

Expert verified
y = -5x + 10

Step by step solution

01

Identify Given Information

The exercise gives the slope of the line, which is denoted by the letter 'm'. Here, the slope is given as m=-5, and the line passes through the point (1, -5).
02

Use Point-Slope Form

The point-slope form of a linear equation is given by y-y_1=m(x-x_1). Use the point provided (1, -5) as (x_1, y_1). Substitute these values and the slope into the point-slope equation: y-(-5)=-5(x-1).
03

Simplify the Equation

Now, simplify the equation: y+5=-5(x-1). Then distribute -5 on the right side: y+5=-5x+5. Combine like terms to get the final equation in slope-intercept form: y=-5x+10.

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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 to write the equation of a line when you know a point on the line and its slope. The general formula is: \[ y - y_1 = m(x - x_1) \] where
  • \( m \) is the slope of the line
  • \( (x_1, y_1) \) is a point on the line.
In our exercise, we are given the slope \( m = -5 \) and the point \( (1, -5) \). Substituting these values into the point-slope formula gives us: \[ y - (-5) = -5(x - 1) \] This equation can be used as it is, but often we simplify it further.
Slope-Intercept Form
The Slope-Intercept form is another popular way to express the equation of a line. It is given by: \[ y = mx + b \] where
  • \( m \) is the slope of the line
  • \( b \) is the y-intercept (the point where the line crosses the y-axis)
After deriving the point-slope form, we often convert it to the slope-intercept form for easier readability. From our point-slope form equation \( y + 5 = -5(x - 1) \), we simplify and distribute to get: \( y = -5x + 10 \). This is our final slope-intercept form.
Simplifying Equations
Simplifying equations means performing algebraic operations to make them more compact and easier to interpret. In our problem, we start with the point-slope form \( y + 5 = -5(x - 1) \). First, let's distribute \( -5 \) on the right side: \[ y + 5 = -5x + 5 \]. Next, subtract \( 5 \) from both sides to isolate \( y \): \[ y = -5x + 10 \]. Now, the equation is in slope-intercept form, showing the slope \( m = -5 \) and the y-intercept \( b = 10 \). Simplifying equations helps us better understand and use them.
Slope of a Line
The slope of a line, commonly denoted by \( m \), represents the steepness and direction of the line. It is calculated as the 'rise' over the 'run' between two points on the line: \[ m = \frac{{y_2 - y_1}}{{x_2 - x_1}} \]. In our given problem, the slope is directly provided as \( m = -5 \). A negative slope indicates that the line descends from left to right. Understanding the slope is crucial as it influences the line's equation and helps in graphing it correctly.

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Most popular questions from this chapter

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