29 Review of Solving Systems of Equations
Instructor’s Notes
One mathematical skill you will need in this unit is the ability to solve systems of linear equations. In class, and on exams, we will stick to two-equations with two-unkowns (what I call 2×2). However, in some of the additional practice problems, you will need to go to three-equations with three-unknowns (3×3). To help everyone refresh their memories on how to do this, I am assigning a few problems. If you are familiar, just go ahead and try them. If you need some review, I include Section 7.1 – Systems of Linear Equations: Two Variables from the OpenStax College Algebra 2e textbook below.
Homework Problems
18. Solve a system with two equations and two unknowns.
19. Solve a system with three equations and three unknowns.
Systems of Linear Equations: Two Variables
Learning Objectives
In this section, you will:
- Solve systems of equations by graphing.
- Solve systems of equations by substitution.
- Solve systems of equations by addition.
- Identify inconsistent systems of equations containing two variables.
- Express the solution of a system of dependent equations containing two variables
A skateboard manufacturer introduces a new line of boards. The manufacturer tracks its costs, which is the amount it spends to produce the boards, and its revenue, which is the amount it earns through sales of its boards. How can the company determine if it is making a profit with its new line? How many skateboards must be produced and sold before a profit is possible? In this section, we will consider linear equations with two variables to answer these and similar questions.
Introduction to Systems of Equations
In order to investigate situations such as that of the skateboard manufacturer, we need to recognize that we are dealing with more than one variable and likely more than one equation. A system of linear equations consists of two or more linear equations made up of two or more variables such that all equations in the system are considered simultaneously. To find the unique solution to a system of linear equations, we must find a numerical value for each variable in the system that will satisfy all equations in the system at the same time. Some linear systems may not have a solution and others may have an infinite number of solutions. In order for a linear system to have a unique solution, there must be at least as many equations as there are variables. Even so, this does not guarantee a unique solution.
In this section, we will look at systems of linear equations in two variables, which consist of two equations that contain two different variables. For example, consider the following system of linear equations in two variables.
The solution to a system of linear equations in two variables is any ordered pair that satisfies each equation independently. In this example, the ordered pair (4, 7) is the solution to the system of linear equations. We can verify the solution by substituting the values