Linear Equation Solution Sets as Graphs

Instructor: Laura Pennington

Laura received her Master's degree in Pure Mathematics from Michigan State University. She has 15 years of experience teaching collegiate mathematics at various institutions.

This lesson will show the relationship between the solution set of a linear equation and the graph of the equation itself. We will look at examples of representing linear equation solution sets as graphs.

Linear Equations and Their Solutions

Suppose you just signed up for a fitness challenge at the gym. The rules are that you get 3 points per hour of working out that you do in the allotted time frame. Whomever has the most points at the end of the challenge wins. We can represent your running total of points with the following equation:

y = 3x

In mathematics, we call this a linear equation in two variables. A linear equation in two variables is a polynomial equation, with a highest exponent of one, that can be put in the form y = mx + b. A solution to a linear equation in two variables consists of values of the variables that make the equation true. We represent solutions with the ordered pair, (x,y). For example, the ordered pair (4,12) is a solution to your fitness equation, because if we plug 4 in for x and 12 in for y, we get a true statement.


solsetlineq1


The solution set of a linear equation in two variables is the set of all of the solutions to the equation. That is, the set of all ordered pairs that satisfy the equation. There are a number of ways to represent a solution set. We can even represent solution sets to linear equations in two variables on a graph, and that is exactly what we are interested in doing in this lesson!

Solution Sets as Graphs

Let's think about how we would represent a solution set to a linear equation in two variables graphically. Hmm…well, we did say that a solution is an ordered pair, and we can plot ordered pairs on a graph! Ah! It's all coming together now; the solution set of a linear equation in two variables can be represented graphically by plotting all of the ordered pair solutions on a graph. Furthermore, this graph would consist of all of the ordered pairs that satisfy the equation, so really, representing the solution set to a linear equation graphically is the exact same thing as graphing the equation itself!

Perfect! To display a solution set graphically, we simply graph the linear equation. Thankfully, these types of equations are one of the simplest to graph. You see, the graph of a linear equation is a line. Because of this, all we need to do is plot a few points that satisfy the equation and then basically just connect the dots in a straight line!

To represent the solution set of a linear equation, y = mx + b, we graph the equation using the following steps:

  1. Find two or more points that satisfy the equation.
  2. Plot those points on a graph.
  3. Connect the points in a straight line.

No problem! Let's give it a try with your fitness equation. First, we find two or more points that satisfy the equation. To do this, we simply plug in a value of x and solve for y to get a solution point. Since we already know the point (4,12) is a solution, let's just find two more, say at x = 1 and at x = 7.


solsetlineq2


Great! We have that (1,3), (4,12) and (7,21) satisfy the equation. The next step is to plot these points on a graph.


solsetlineq3


Almost done! Last step is to connect the dots with a straight line.


solsetlineq4


Pretty easy! We've graphed our equation, y = 3x, and in doing so, we've represented the solution set of the equation graphically!

Another Example

This representing of solution sets graphically isn't so bad! To really solidify our understanding, let's try one more example. Suppose you are renting a speedboat, and the rental cost is such that there is a $20 base fee plus $25 per hour. Therefore, the rental cost can be represented using the following linear equation:

y = 25x + 20

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