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Chapter 5: Problem 24

Write an equation of the line satisfying the given conditions. Passing through \((0,-2)\) and \((-2,0)\)

Short Answer

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

Step by step solution

01

Identify the coordinates

The given points are \(0, -2\) and \(-2, 0\). Let's call them \(x_1, y_1\) and \(x_2, y_2\). So, \(x_1 = 0\), \(y_1 = -2\), \(x_2 = -2\), and \(y_2 = 0\).
02

Calculate the slope (m)

Use the slope formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] Substituting in the coordinates: \[ m = \frac{0 - (-2)}{-2 - 0} = \frac{2}{-2} = -1 \]
03

Use the point-slope form

Now insert the slope \(m = -1\) and one of the points, let's use \( (0, -2) \), into the point-slope form equation: \[ y - y_1 = m(x - x_1) \] This gives: \[ y - (-2) = -1(x - 0) \]
04

Simplify to slope-intercept form

Simplify this equation to get it into slope-intercept form \(y = mx + b\): \[ y + 2 = -x \] Subtract 2 from both sides to solve for y: \[ y = -x - 2 \]
05

Conclusion

The equation of the line passing through the points \(0,-2\) and \(-2,0\) is \[ y = -x - 2 \]

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Key Concepts

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

slope
The slope of a line measures its steepness and direction. It is calculated as the change in the y-coordinates divided by the change in the x-coordinates between two points. This ratio tells us how much y changes per unit change in x. The formula for the slope (m) is: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] For the given points (0, -2) and (-2, 0), we calculated: \[ m = \frac{0 - (-2)}{-2 - 0} = \frac{2}{-2} = -1 \] This means the line decreases by 1 unit vertically for every 1 unit it moves horizontally to the right.
point-slope form
The point-slope form of a line's equation is a very convenient way to write the equation when you know the slope and one point on the line. The general formula is: \[ y - y_1 = m(x - x_1) \] In this format,
  • (y - y_1) shifts the line vertically by y_1.
  • (x - x_1) shifts the line horizontally by x_1.
  • The slope (m) is the same as discussed previously.
For our example with the slope -1 and point (0, -2), the equation is: \[ y - (-2) = -1(x - 0) \] Which simplifies to: \[ y + 2 = -x \]
slope-intercept form
The slope-intercept form of a line's equation is quite popular and easy to use. It looks like this: \[ y = mx + b \] Here, m is the slope of the line, and b is the y-intercept (the point where the line crosses the y-axis). In our example, after converting from the point-slope form, we got: \[ y + 2 = -x \] To convert this into slope-intercept form, subtract 2 from both sides: \[ y = -x - 2 \] This tells us that the slope (m) is -1 and the y-intercept (b) is -2. This means the line crosses the y-axis at -2.
coordinates
Coordinates are used to locate points on the Cartesian plane. They consist of an ordered pair (x, y), where:
  • The x-coordinate tells you how far to move left or right from the origin (0, 0).
  • The y-coordinate tells you how far to move up or down from the origin.
For example, in our exercise, the points (0, -2) and (-2, 0) have:
  • (0, -2): Start at the origin, move 0 units left/right (x=0), and 2 units down (y=-2).
  • (-2, 0): Start at the origin, move 2 units left (x=-2), and 0 units up/down (y=0).
Understanding the coordinates helps in plotting points and grasping the geometric representation of equations on the graph.

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Most popular questions from this chapter

Round off to the nearest hundredth when necessary. A sidewalk food vendor knows that selling 80 franks costs a total of 73 dollar and selling 100 franks costs a total of 80 dollar Assume that the total cost \(c\) of the franks is linearly related to the number \(f\) of franks sold. (a) Write an equation relating the total cost of the franks to the number of franks sold. (b) Find the cost of selling 90 franks. (c) How many franks can be sold for a total cost of 90.50 dollar (d) The costs that exist even if no items are sold are called the fixed costs. Find the vendor's fixed costs. I Hint: If no franks are sold then \(f=0.1\)

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Write an equation of the line satisfying the given conditions. Line has \(y\) -intercept 3 and \(x\) -intercept 7

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Determine the slope of the line from its equation. $$y=-4 x+2$$
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