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Chapter 7: Problem 68

Find the slope of each line, and sketch its graph. $$ y=-3 x $$

Short Answer

Expert verified
The slope is -3. Plot points like (0,0) and (1,-3) and draw the line through them.

Step by step solution

01

Identify the slope in the given equation

The provided equation is in the slope-intercept form, which is given by y = mx where m is the slope of the line. Identify the slope (m) from the equation y = -3x. In this case, m = -3.
02

Understand the meaning of the slope

The slope m = -3 indicates how steep the line is and the direction it moves. A slope of -3 means that for every 1 unit increase in x, y decreases by 3 units.
03

Plot points using the slope

To sketch the graph, start at the origin (0, 0). Use the slope to find another point. For example, starting from (0, 0), move 1 unit to the right (increase x by 1) and then move 3 units down (decrease y by 3) to get the point (1, -3).
04

Draw the line

Plot the points obtained and use a ruler to draw a straight line through these points extending in both directions. This line represents the graph of y = -3x.

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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 an essential concept in algebra. It is written as:
\[ y = mx + b \]
In this equation, m represents the slope of the line, and b is the y-intercept (the point where the line crosses the y-axis).
For example, in the equation y = -3x, the slope m is -3 and the y-intercept b is 0.

When the y-intercept is 0, the equation can be simplified further to: \[ y = mx \]

Understanding the slope-intercept form makes it easier to graph linear equations, identify slopes, and calculate where the line intersects the y-axis.
To summarize:
  • m (the slope) describes the line's steepness and direction
  • b (the y-intercept) indicates where the line crosses the y-axis
graphing linear equations
Graphing linear equations involves plotting points on a coordinate plane and drawing a line through these points.
When graphing an equation in slope-intercept form (such as y = -3x), the slope and y-intercept provide crucial information.
Here's how to graph a linear equation:
  • Identify the slope (m) and the y-intercept (b) from the equation.
  • Start by plotting the y-intercept on the y-axis. In our example, b = 0, so the first point is (0, 0).
  • Use the slope to find additional points. A slope of -3 means that for every 1 unit increase in x, y decreases by 3 units.
  • Plot these additional points. From (0, 0), move 1 unit to the right (increase x by 1) and 3 units down (decrease y by 3) to get another point like (1, -3).
  • Draw a straight line through all the points.
That's it! Graphing linear equations helps visualize their behavior and relationships.
negative slope
A negative slope means that as the x value increases, the y value decreases. In other words, the line slopes downwards from left to right.
Consider the equation y = -3x again. Here, the slope m is -3.
This tells us that for every 1 unit increase in x, the y value will decrease by 3 units.
This concept is crucial for understanding the direction and angle of a line.
  • If the slope was positive, the line would slope upwards from left to right.
  • A larger absolute value of the negative slope means a steeper decline.
So, visualizing a negative slope helps understand how linear equations behave and how changes in x affect y. This is why graphing practice is useful 鈥� it brings these concepts to life!

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Most popular questions from this chapter

How many pounds of candy that sells for \(\$ 2.50\) per \(1 \mathrm{~b}\) must be mixed with candy that sells for \(\$ 1.75\) per \(\mathrm{lb}\) to obtain \(6 \mathrm{lb}\) of a mixture that sells for \(\$ 2.10\) per lb?

Braving blizzard conditions on the planet Hoth, Luke Skywalker sets out in his snow speeder for a rebel base \(4800 \mathrm{mi}\) away. He travels into a steady headwind and makes the trip in \(3 \mathrm{hr}\). Returning, he finds that the trip back, now with a tailwind, takes only \(2 \mathrm{hr}\). Find the rate of Luke's snow speeder and the wind speed.

Without graphing, do the following for each system of equations. (a) Describe each system. (b) State the number of solutions. (c) Is the system inconsistent, are the equations dependent, or neither? \(y+2 x=6\) \(x-3 y=-4\)

Solve each system by graphing. If the system is inconsistent or the equations are dependent, say so. \(2 x-y=6\) \(4 x-2 y=8\)

A train travels \(150 \mathrm{~km}\) in the same time that a plane travels \(400 \mathrm{~km}\). If the rate of the plane is \(20 \mathrm{~km}\) per hr less than three times the rate of the train, find both rates.
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