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Chapter 1: Problem 19

Find an equation of the given line. Slope is \(-2 ; x\)-intercept is \(-2\)

Short Answer

Expert verified
The equation of the line is \( y = -2x - 4 \).

Step by step solution

01

Understand the 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 Values

We are given the slope \( m = -2 \) and the x-intercept \( -2 \).
03

Find a Point on the Line

Since the x-intercept is \( -2 \), it means the line passes through the point \( (-2, 0) \).
04

Use the Point-Slope Formula

The point-slope form of the equation of a line is given by: \( y - y_1 = m(x - x_1) \)Substitute \( m = -2 \), \( x_1 = -2 \) and \( y_1 = 0 \).
05

Substitute and Simplify

Plug in the values: \[ y - 0 = -2(x - (-2)) \]Simplify: \[ y = -2(x + 2) \]Distribute the \( -2 \):\[ y = -2x - 4 \]

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Key Concepts

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

Slope-Intercept Form
The slope-intercept form is an essential concept in understanding linear equations. The general formula is:
\(y = mx + b\)
Here, \(m\) represents the slope, and \(b\) is the y-intercept where the line crosses the y-axis.
To easily graph a line, identify the slope and y-intercept from this equation.
  • The slope \(m\) tells you how steep the line is and the direction it goes.
  • The y-intercept \(b\) shows where the line will cross the y-axis.
For example, in the equation \(y = -2x - 4\), the slope \(m\) is \(-2\), and the y-intercept \(b\) is \(-4\).
This means the line goes down 2 units for every 1 unit it goes right, and it crosses the y-axis at the point (0, -4).
Point-Slope Form
The point-slope form is another way to represent the equation of a line. It's especially useful if you know a point on the line and the slope. The formula is:
\(y - y_1 = m(x - x_1)\)
Here, \(m\) is the slope and \((x_1, y_1)\) is a specific point on the line.
This form allows you to find the equation quickly.
  • First, substitute the slope \(m = -2\) and the point \((-2, 0)\) into the formula.
  • This gives you \(y - 0 = -2(x - (-2))\).
  • Simplify this to get the equation in slope-intercept form.
For our example, simplifying yields:
\(y = -2(x + 2)\), and then \(y = -2x - 4\).
This transformation shows how information from one form can convert into another, making it versatile for various math problems.
X-Intercept
The x-intercept of a line is the point where it crosses the x-axis. At this point, the value of \(y\) is zero. Finding the x-intercept is straightforward:
  • Set \(y = 0\) in the equation and solve for \(x\).
If an equation is given in slope-intercept form, for instance, \(y = -2x - 4\):
\[0 = -2x - 4\]
Add 4 to both sides:
\[4 = -2x\]
Divide by -2:
\(x = -2\)
This means the x-intercept is -2, confirming that the line crosses the x-axis at (-2, 0).
Understanding x-intercepts helps when graphing lines and analyzing where lines intersect axes.

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Most popular questions from this chapter

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