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Chapter 1: Problem 5

Graph. Find the slope and the \(y\) -intercept. \(y=-3 x\)

Short Answer

Expert verified
Slope: -3, y-intercept: 0.

Step by step solution

01

Identify the Equation Form

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

Determine the Slope \( m \)

In the equation \( y = -3x \), compare it to the slope-intercept form \( y = mx + b \). The coefficient of \( x \) is the slope \( m \). Here, \( m = -3 \).
03

Determine the \( y \)-intercept \( b \)

In the equation \( y = -3x \), there is no constant term added or subtracted, implying \( b = 0 \). Therefore, the \( y \)-intercept is \( 0 \).
04

Summary

The slope of the given line is \( -3 \) and the \( y \)-intercept is \( 0 \).

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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 measures its steepness and direction. It indicates how much the value of y changes for each unit increase in the value of x. Typically, slope is represented by the letter 'm'.

In the equation of a line in slope-intercept form, which is written as \(y = mx + b\), the slope is the coefficient of the x-term.

For example, in the equation \(y = -3x\), the coefficient of x is -3, so the slope \(m = -3\).

A positive slope means the line goes upwards as you move from left to right.
A negative slope means the line goes downwards as you move from left to right.
A larger positive or negative value of slope means a steeper line.
y-intercept
The y-intercept is the point where the line crosses the y-axis. It represents the value of y when x is zero. In other words, this is where the line touches or cuts the y-axis.

In the equation \(y = mx + b\), the y-intercept is given by the constant term 'b'.

For instance, in the equation \(y = -3x\), there isn鈥檛 an additional constant term, meaning the y-intercept, \(b = 0\). So, the line crosses the y-axis at the origin (0, 0).
  • If the y-intercept is a positive number, the line crosses the y-axis above the origin.
  • If it's negative, the line crosses below the origin.
Linear Equation
A linear equation is any equation that, when graphed, gives a straight line. Linear equations can be written in many forms, but one of the most common is the slope-intercept form: \(y = mx + b\).

In the slope-intercept form:
  • 'y' is the dependent variable.
  • 'x' is the independent variable.
  • 'm' is the slope of the line.
  • 'b' is the y-intercept, where the line meets the y-axis.

For example, the equation \(y = -3x\) depicts a straight line with a slope of -3 and a y-intercept of 0. When graphed, this line would go downwards steeply since the slope is negative and the line passes through the origin.

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