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Chapter 3: Problem 15

Use the slope-intercept form to graph each equation. See Examples 2 and 3. $$ y=-5 x $$

Short Answer

Expert verified
Graph the line passing through the origin and point (1,-5) with a slope of -5.

Step by step solution

01

Identify the equation form

The given equation is in the form \( y = mx + b \), which is the slope-intercept form, where \( m \) is the slope and \( b \) is the y-intercept. The equation given is \( y = -5x \).
02

Determine the slope

In the equation \( y = -5x \), the slope \( m \) is \(-5\). This means for every 1 unit increase in \( x \), \( y \) decreases by 5 units.
03

Identify the y-intercept

The y-intercept \( b \) is the constant term in the equation. Here, \( b = 0 \) since there is no constant term added to \(-5x\), which means the line crosses the y-axis at the origin (0,0).
04

Plot the y-intercept

On a coordinate grid, plot the y-intercept (0,0), which is where the line will cross the y-axis.
05

Use the slope to find another point

Starting from the y-intercept (0,0), apply the slope \(-5\). Move right 1 unit to increase \( x \) by 1, and move down 5 units to decrease \( y \) by 5. This gives the second point (1, -5).
06

Draw the line

Connect the points (0,0) and (1,-5) with a straight line extending in both directions. This line represents the graph of the equation \( y = -5x \).

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Key Concepts

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

Graphing Linear Equations
When it comes to graphing linear equations, the slope-intercept form is a handy tool. This form is represented as \( y = mx + b \), where \( m \) is the slope, and \( b \) is the y-intercept. To graph an equation like \( y = -5x \), we start by identifying these two key components.
  • First, recognize that the equation is already in slope-intercept form.
  • Next, identify the slope \(-5\) and the y-intercept \(b = 0\).
  • Plot the y-intercept on the graph. In this case, the point is \( (0,0) \).
This point marks where the line will cross the y-axis. From the y-intercept, use the slope to find additional points. Simply move according to the slope's instructions: right and down if the slope is negative, or right and up if positive. Connect these points with a straight line, ensuring it extends infinitely in both directions. That's your linear graph!
Slope
The concept of slope describes the steepness and direction of a line, and it can significantly aid in graphing linear equations. The slope \( m \) in the equation \( y = mx + b \) tells us how much \( y \) changes with a unit change in \( x \). For the equation \( y = -5x \), the slope is \(-5\).
  • A slope of \(-5\) means that for every unit you move to the right on the x-axis, the y-value decreases by 5 units.
  • This consistent rate of change determines the angle and the direction of the line on a graph.
A negative slope like \(-5\) causes the line to slope downwards from left to right. Interpreting a slope is crucial as it guides you in choosing points to accurately draw a line on the coordinate grid. To find another point from the y-intercept, count upwards or downwards along the y-axis based on the slope, plotting these positions to graph the line.
Y-intercept
The y-intercept \( b \) in the slope-intercept form \( y = mx + b \) indicates where the line crosses the y-axis. In our example, \( y = -5x \), there is no separate constant term added or subtracted 鈥� hence, the y-intercept is 0.
  • The y-intercept is crucial because it is one of the easiest points to plot on a graph. For this equation, it is the origin \( (0,0) \).
  • Once plotted, it serves as a starting point for charting additional points based on the slope's direction.
Having identified the y-intercept simplifies the process of plotting a linear equation because you have a guaranteed point to work from. Always start by plotting the y-intercept and then use the slope to determine further coordinates, ensuring the line is drawn accurately.

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Most popular questions from this chapter

Solve. Assume each exercise describes a linear relationship. Write the equations in slope-intercept form. See Example 8 . In 2006 , the U.S. population per square mile of land area was \(85 .\) In 2000 , the person per square mile population was 79.6 a. Write an equation describing the relationship between year and persons per square mile. Use ordered pairs of the form (years past \(2000,\) persons per square mile). b. Use this equation to predict the person per square mile population in \(2010 .\)

Find the slope of each line. \(-4 x-7 y=9\)

\(10+3 x=5 y\) \(5 x+3 y=1\)

See Examples 1 through 7 . Find an equation of each line described. Write each equation in slope-intercept form (solved for \(y\) ), when possible. With slope 0, through \((6.7,12.1)\)

Find the value of \(x^{2}-3 x+1\) for each given value of \(x .\) See Section 1.7. $$ 5 $$
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