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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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