Unit 5: Graphs of Linear Equations and Inequalities
We use graphs to help us visualize how one quantity relates to another. This unit will help you become comfortable with graphing pairs of numbers on the coordinate plane and understand how we can use lines to represent equations and relationships.
For example, we can graph how the location of a train depends on when it left the station. If the train is moving at constant speed, the line in the graph is straight. The slope or slant of the line depends on the speed: the greater the speed, the steeper the line. If the line is going up (from left to right), it tells you the distance is growing with time: the train is moving away from the station. If the line is going down, it tells you the distance is decreasing: the train is approaching the station. You can gather a lot of information about the train's journey from just one graph.
Completing this unit should take you approximately 5 hours.
Upon successful completion of this unit, you will be able to:
- graph points with given coordinates on the rectangular coordinate plane;
- determine coordinates of a point on the rectangular coordinate system;
- determine whether a given ordered pair is a solution of the equation with two variables;
- find and graph solutions of the equation in two variables;
- graph a straight line given either its equation, or a slope and y-intercept;
- find slope and intercepts of a straight line given its equation or its graph;
- write the equation of a line passing through two given points;
- write the equation of a line with a given slope passing through a given point;
- locate on a coordinate plane all solutions of a given inequality in two variables;
- represent relationships between quantities as an equation or inequality in two variables; and
- interpret the meaning of slope and intercepts of the graph of a relationship between quantities.
5.1: Graphing Points in the Rectangular Coordinate Plane
First, we need to understand the coordinate plane, the space in which we produce graphs. In this section we will focus on finding and graphing points on the coordinate plane to become comfortable with it.
Read this page and watch the videos in the text. Pay attention to the example near the end of the page and how the author identifies points that do not directly match up with the tick marks on the graph.
5.2: Ordered Pairs as Solutions of an Equation in Two Variables
Now we are ready to begin using graphs to determine if a pair of numbers (an ordered pair) is a solution to an equation.
In this video, we see that if we graph an equation, the solutions must lie on the graph. This gives us an additional way to test if an ordered pair of numbers is a solution to an equation.
After you watch, complete this assessment and choose to solve the equations using algebra or graphs.
5.3: Graphing Equations in Two Variables of the Form Ax + By = C
A common way equations can be written is: Ax + By = C, where A, B, and C are numbers.
Watch these videos, which explain how to manipulate an equation into a format that allows for graphing.
5.4: Intercepts of a Straight Line
One of the properties of linear graphs is that they have intercepts on the x- and y-axis. The intercept is the point at which the line crosses the axis.
Read this section to learn how to identify intercepts and how to use them in graphing and the general form of x- and y-intercepts.
After you read, complete examples 4.19 through 4.24.
5.5: Definition of Slope and Slope Formula
Another important property of linear graphs is the slope of the graph. The slope tells us how steep the line is. When a linear equation is written in a specific form that we’ll discuss later, the slope helps us determine how to graph the line.
Read this section. Pay attention to the definition of slope and how we define positive and negative slope. Also, pay attention to the slope formula given toward the end of the page. The slope formula will allow you to calculate the slope of any given line.
After you read, complete examples 4.25, 4.26, and 4.29 through 4.37 and check your work.
5.6: Slopes of Parallel and Perpendicular Lines
Here, we learn about how the slopes of parallel and perpendicular lines are related. Parallel lines have the same slope, while perpendicular lines have slopes that are reciprocals.
Read this page to see the special relationship of the slopes of parallel and perpendicular lines. After you read, try a few practice problems.
5.7: Graphing Equations in Two Variables of the Form y = mx + b
One of the most common types of graph is that of a line with the form y = mx + b. In this form, m is the slope of the line, and b is the y-intercept of the line. When an equation is in this form, it is easy to plot the linear graph, so it is important to be able to recognize when an equation is in this form.
Read up to the section on choosing the most convenient method to graph a line. The form y = mx + b is often called the slope-intercept form of a linear equation.
After you read, complete examples 4.40 through 4.46 and check your work.
5.8: Point-Slope Form
In the last section we discussed the slope-intercept form of a linear equation. We can also write linear equations in a form known as the point-slope form. This form works for when you want to make a line between two known points.
This form is: y − y1 = m(x − x1).
Read this page, which describes writing and graphing equations in point-slope and shows how to convert from the point-slope form to slope-intercept form, which can often be more useful. Watch the videos for more examples.
After you read, complete review problems 3, 4, 7, 8, 14, and 15 and check your answers.
5.9: Graphing Linear Inequality of Two Variables on the Coordinate Plane
The last type of linear graphing we need to study is the graph of an inequality rather than an equation. When we graph inequalities, we must pay attention not only to the numbers and variables but also the inequality itself. That is, are we graphing a less-than, or greater-than inequality?
Read this page, which reviews some calculations with inequalities and shows how we use shading on a graph to indicate inequality. Pay attention to the summary of graphing inequalities.
Complete examples 4.70 to 4.75 and check your answers.