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Chapter 2: Problem 3

Find the slope and \(y\) -intercept of the graph of the equation \(y=-\frac{2}{3} x+1\)

Short Answer

Expert verified
Slope is \(-\frac{2}{3}\), y-intercept is 1.

Step by step solution

01

Identify the Equation Format

The given equation is in the slope-intercept form, which is expressed as \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.
02

Extract the Slope

In the equation \(y = -\frac{2}{3}x + 1\), the coefficient of \(x\) is \(-\frac{2}{3}\). This value represents the slope \(m\).
03

Extract the Y-intercept

The constant term in the equation, which is \(+1\), represents the y-intercept \(b\).
04

State the Slope and Y-intercept

The slope \(m\) is \(-\frac{2}{3}\), and the y-intercept \(b\) is 1.

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Key Concepts

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

Linear Equations
A linear equation describes a straight line on a Cartesian plane. The standard form involves expressions where each term is either a constant or the product of a constant and a single variable. This form is usually written as \(Ax + By = C\), but the slope-intercept form is more user-friendly for graphing purposes.

In the slope-intercept form, the equation is expressed as \(y = mx + b\), where:
  • \(y\) is the dependent variable (usually the vertical axis).
  • \(x\) is the independent variable (usually the horizontal axis).
  • \(m\) is the slope of the line.
  • \(b\) is the y-intercept, where the line crosses the y-axis.
This form allows you to quickly determine where the line crosses the y-axis and how steep it is, providing a quick glance at the equation's graphical representation.
Slope
The slope of a line is a measure of its steepness. It tells you how much \(y\) changes for a given change in \(x\). The slope is represented by \(m\) in the slope-intercept equation \(y = mx + b\).

The slope can be understood as follows:
  • If \(m > 0\), the line is increasing (sloping upwards).
  • If \(m < 0\), the line is decreasing (sloping downwards).
  • If \(m = 0\), the line is horizontal, indicating constant \(y\) values.
In the example equation \(y = -\frac{2}{3}x + 1\), the slope is \(-\frac{2}{3}\). This means for every 3 units you move to the right along the x-axis, the value of \(y\) decreases by 2 units. This negative value indicates a decreasing line.
Y-Intercept
The y-intercept is a crucial component of the linear equation, highlighting where the line crosses the y-axis. In the slope-intercept form \(y = mx + b\), the y-intercept is represented by \(b\).

The y-intercept can be understood through the following:
  • The point where the line intersects the y-axis.
  • This point has coordinates \((0, b)\).
For our provided equation \(y = -\frac{2}{3}x + 1\), the y-intercept is 1. This tells you that, when \(x = 0\), \(y = 1\). Hence, the line crosses the y-axis at the point \((0, 1)\). Knowing the y-intercept helps in quickly sketching the line on a graph and aids in understanding the graph's starting point in a real-world context.

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Most popular questions from this chapter

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