Problem 24 - Understanding Elementary Algebra with Geometry

Question

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

Step-by-Step Solution

Answer

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

Step 1: 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 \(y_2 = 0\).

Step 2: Calculate the slope (m)

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

Step 3: 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: \[ y - (-2) = -1(x - 0) \]

Step 4: 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: \[ y = -x - 2 \]

Step 5: Conclusion

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

Key Concepts

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.