/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 3 The graph of \(y=3 x-7\) represe... [FREE SOLUTION] | 91影视

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Chapter 3: Problem 3

The graph of \(y=3 x-7\) represents all solutions of the equation.

Short Answer

Expert verified
Plot the y-intercept at (0, -7), use the slope to find (1, -4), and draw a line through these points.

Step by step solution

01

Identify the Equation Given

First, recognize the equation given in the problem, which is the linear equation: \(y = 3x - 7\)
02

Understand the Linear Equation

This equation is in the slope-intercept form, \(y = mx + b\), where \(m = 3\) is the slope, and \(b = -7\) is the y-intercept. This means that the graph will be a straight line crossing the y-axis at \(-7\) and having a slope of \(3\).
03

Plot the Y-Intercept

On a coordinate plane, locate the y-intercept. From the equation, the y-intercept \(b = -7\). This means the line will cross the y-axis at the point \((0, -7)\). Plot this point on the graph.
04

Use the Slope to Find Another Point

The slope \(m = 3\) can be interpreted as 'rise over run', or \(3/1\). Starting from the y-intercept at \((0, -7)\), move up 3 units and 1 unit to the right to find another point on the line. This point will be at \((1, -4)\). Plot this point on the graph.
05

Draw the Line

Using a ruler, draw a straight line through the two points: \((0, -7)\) and \((1, -4)\). This line represents the graph of the equation \(y = 3x - 7\), and it includes all solutions to the equation.

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Key Concepts

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

Slope-Intercept Form
The slope-intercept form of a linear equation is a simple way to graph linear functions. It follows the formula: \(y = mx + b\). Here, \(m\) is the slope, and \(b\) is the y-intercept.
The equation \(y = 3x - 7\) follows this format perfectly.
The slope-intercept form gives you instant information about the line, making it easy to graph.
  • You can identify the slope directly.
  • You can find the y-intercept and use it to start plotting your graph.

If you memorize this form, it'll be much easier to quickly understand and sketch any linear equation.
Y-Intercept
The y-intercept is the point where the line crosses the y-axis. This occurs when \(x = 0\).
For the equation \(y = 3x - 7\), we can identify the y-intercept as \(b = -7\).
  • This means the line will cross the y-axis at the point \((0, -7)\).
  • Always start by plotting this point first when you're drawing the line.

By plotting the y-intercept first, you establish the initial point of your line, making it straightforward to use the slope to find additional points.
Slope
The slope indicates both the direction and steepness of a line. It's calculated as 'rise over run', showing how much the line goes up or down for each unit it moves horizontally.
For the equation \(y = 3x - 7\), the slope \(m = 3\).
This means for every unit you move to the right along the x-axis, you move up three units on the y-axis.
  • If the slope were negative, the line would move downwards instead.
  • You can use the slope to locate other points on the line by starting at the y-intercept and moving according to the slope's 'rise over run'.

For a slope of 3, start at the y-intercept \((0, -7)\), go up 3 units, and 1 unit to the right to find the next point \((1, -4)\). Connecting these points will give you the full graph of the line.

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Most popular questions from this chapter

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