Problem 20

Question

Exer. 19-20: Sketch the graphs of the lines on the same coordinate plane. $$ y=-2 x-1, \quad y=-2 x+3, \quad y=\frac{1}{2} x+3 $$

Step-by-Step Solution

Verified

Answer

Lines A and B are parallel and will never intersect, while Line C intersects Line B at $(0, 3)$.

1Step 1: Identify the Line Equations

We have three equations of lines to sketch: 1. Line A: $ y = -2x - 1 $2. Line B: $ y = -2x + 3 $3. Line C: $ y = \frac{1}{2}x + 3 $.Each one is in the form $ y = mx + b $, where $ m $ is the slope and $ b $ is the y-intercept.

2Step 2: Determine the Slopes and Intercepts

From the equations, we identify:- For Line A, slope $ m = -2 $ and y-intercept $(0, -1)$.- For Line B, slope $ m = -2 $ and y-intercept $(0, 3)$.- For Line C, slope $ m = \frac{1}{2} $ and y-intercept $(0, 3)$.

3Step 3: Plot the Y-Intercepts

On a coordinate plane, plot the y-intercepts:- Line A intercepts the y-axis at $ (0, -1) $.- Line B intercepts the y-axis at $ (0, 3) $.- Line C also intercepts the y-axis at $ (0, 3) $.

4Step 4: Use the Slopes to Plot Additional Points

For each line, use the slope to find another point:- Line A: With a slope of $-2$, move down 2 units and right 1 unit from $(0, -1)$ to $(1, -3)$.- Line B: With a slope of $-2$, move down 2 units and right 1 unit from $(0, 3)$ to $(1, 1)$.- Line C: With a slope of $\frac{1}{2}$, move up 1 unit and right 2 units from $(0, 3)$ to $(2, 4)$.

5Step 5: Draw the Lines

Draw lines through the points plotted for each equation:- Line A through $(0, -1)$ and $(1, -3)$.- Line B through $(0, 3)$ and $(1, 1)$.- Line C through $(0, 3)$ and $(2, 4)$.

Key Concepts

Coordinate GeometrySlope-Intercept FormSlope of a LineY-Intercept

Coordinate Geometry

Coordinate geometry, often referred to as analytic geometry, is a branch of mathematics that uses algebra to study geometric problems. It allows the representation of geometric shapes in a numerical way through the use of a coordinate plane. The coordinate plane is a two-dimensional surface in which each point is defined by a pair of numerical coordinates. These coordinates are written in the form

(x, y)

where x represents the horizontal position and y the vertical position. This system allows for the visualization and analysis of geometric shapes and lines. By plotting lines on a coordinate plane, you can see how they intersect and relate to each other, making it a crucial tool for understanding various properties of shapes in a more concrete manner.

Slope-Intercept Form

The slope-intercept form is an essential concept in understanding linear equations and is expressed as

$ y = mx + b $

This formula is a way to write the equation of a line, where:

$ m $ represents the slope of the line.
$ b $ denotes the y-intercept.

The slope

$ m $

is a measure of how steep a line is and indicates the rate at which y changes with respect to x. The y-intercept

$ b $

represents the point where the line crosses the y-axis, providing a starting point for the line on the graph. By using the slope-intercept form, it becomes straightforward to graph a line and predict its behavior across the plane.

Slope of a Line

The slope of a line is a critical concept that describes the direction and steepness of the line. It is defined as the change in vertical distance divided by the change in horizontal distance between two points on the line. This ratio is often symbolized as

$ m = \frac{\Delta y}{\Delta x} $

where

$ \Delta y $ is the change in the y-coordinates.
$ \Delta x $ is the change in the x-coordinates.

When the slope is positive, this means the line inclines upwards as it moves from left to right. Conversely, a negative slope indicates that the line declines. A zero slope indicates a horizontal line, and an undefined slope indicates a vertical line. Understanding the slope helps in predicting how the line behaves as it extends across the coordinate plane.

Y-Intercept

The y-intercept is the point where a line crosses the y-axis on a coordinate plane. This occurs when the value of

x is zero.

In a graph of a linear equation, the y-intercept provides a starting position for plotting the line. It helps to understand where the line will emerge on the graph before considering the slope's effect. In the slope-intercept form of linear equations