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Chapter 0: Problem 14

Graph. List the slope and \(y\) -intercept. $$y=2 x-5$$

Short Answer

Expert verified
Slope: 2, y-intercept: -5

Step by step solution

01

Identify the slope-intercept form

Recognize that the equation given, \( y = 2x - 5 \), is in slope-intercept form, which is \( y = mx + b \). Here, \( m \) represents the slope and \( b \) represents the y-intercept.
02

Determine the slope

In the equation \( y = 2x - 5 \), the coefficient of \( x \) is 2. Therefore, the slope \( m \) is 2.
03

Find the y-intercept

In the equation \( y = 2x - 5 \), the constant term is -5. Therefore, the y-intercept \( b \) is -5.
04

Summarize the results

The slope of the equation \( y = 2x - 5 \) is 2, and the y-intercept is -5.

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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 is a measure of how steep the line is. It represents the rate at which the y-coordinate of a point on the line changes as the x-coordinate changes. The slope is often denoted by the letter 'm'. In a linear equation of the form \( y = mx + b \), the coefficient of x is the slope. Think of the slope as the 'rise over run', which is the change in y divided by the change in x. For example, in the equation \( y = 2x - 5 \), the slope is 2. This means that for every 1 unit increase in x, y increases by 2 units. A positive slope means the line goes uphill, while a negative slope means it goes downhill.
Y-Intercept
The y-intercept of a line is the y-coordinate at which the line crosses the y-axis. This is where the value of x is 0. In the slope-intercept form of a linear equation \( y = mx + b \), the y-intercept is represented by 'b'. For example, in the equation \( y = 2x - 5 \), the y-intercept is -5. This means the line crosses the y-axis at the point (0, -5). Knowing the y-intercept is useful for quickly graphing the line, because it gives you a starting point from which you can use the slope to find other points on the line.
Linear Equations
A linear equation is any equation that can be written in the form \( y = mx + b \), where 'm' is the slope and 'b' is the y-intercept. These equations graph as straight lines. Linear equations can model relationships with a constant rate of change. For example, in the equation \( y = 2x - 5 \), y changes by 2 units for every 1 unit change in x. Linear equations are fundamental in algebra and are widely used to solve real-world problems where relationships between quantities are linear.
Graphing
Graphing a linear equation involves plotting points on a coordinate plane and then drawing a line through these points. To graph the equation \( y = 2x - 5 \), you can start by plotting the y-intercept (0, -5). Next, use the slope to find another point. Since the slope is 2, for each step to the right (increasing x by 1), move up 2 units (increasing y by 2). Plot this second point. Continue this process to find more points if needed. Finally, draw a straight line through these points. Graphing helps visualize the relationship described by the equation and can aid in interpreting solutions and understanding trends.

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