Problem 62 - Algebra 1

Question

use the following linear equations. line a: \(y=-x+2\) line \(b: y=-1\) line \(c: y=x+2\) Which of the lines are parallel? Explain.

Step-by-Step Solution

Answer

None of the given lines are parallel as they do not have the same slope.

Step 1: Identify the Slopes

For the equation of a line in the form \(y = mx + c\), 'm' represents the slope. From the given equations, we can identify the slopes as follows:1. For the first line, \(y = -x + 2\), the slope is -1. 2. For the second line, \(y = -1\), it's a horizontal line so the slope is 0.3. For the third line, \(y = x + 2\), the slope is 1.

Step 2: Compare the Slopes

Comparing the slopes identified, we see that none of the given lines have the same slope. Therefore, none of the lines are parallel.

Key Concepts

Linear Equations

Linear equations are the simplest type of equations in algebra that represent straight lines when graphed. They have the general form of y = mx + c, where m is the slope of the line and c is the y-intercept, which is the point where the line crosses the y-axis. In the context of the exercise, we are given three different equations: y=-x+2 (line a), y=-1 (line b), and y=x+2 (line c). These equations can be graphed to depict lines on a two-dimensional plane where the x-axis represents the horizontal direction and the y-axis represents the vertical direction.

Slope is an essential concept in understanding linear equations because it describes the direction and the steepness of a line. The y-intercept is also crucial as it gives a starting point to draw the line on the graph. By analyzing these two components, one can determine if lines are parallel, intersecting, or coinciding.

Slope of a Line

The slope of a line is a measure of its steepness and direction, usually represented by the letter m in a linear equation. It is calculated as the 'rise' over the 'run,' indicating how much the line goes up or down for each unit it goes right or left. A positive slope means the line is inclined upwards, a negative slope means it’s inclined downwards, and a zero slope indicates a horizontal line.

In our example, line a has a slope of -1, indicating it tilts downwards from left to right. Line b, with a slope of 0, is horizontal, meaning it doesn't tilt at all—it's flat. Line c has a slope of 1, which means it inclines upwards from left to right. Understanding slopes is vital because it tells us if two lines are parallel (they will never intersect) or if they will meet at some point.

Comparing Slopes

Comparing slopes of different lines helps us to determine their relationships. If two lines never intersect, they are considered parallel. Parallel lines have the same slope, implying they are moving in the same direction with the same steepness. In contrast, lines that have different slopes will ultimately cross each other at some point.

From the exercise, by comparing the slopes of the three given lines, we noticed that none of them share the same slope value. Line a has a slope of -1, line b has a slope of 0, which is unique to horizontal lines, and line c has a slope of 1. This means that none of the lines are parallel to each other, as they don't have identical slopes. This concept of slope comparison is a reliable method of testing parallelism between lines on the same plane.