Solving Systems of Linear Equations

What is a System of Linear Equations?

A system of linear equations is simply a collection of two or more linear equations that share the same variables. A "solution" to such a system is a set of values for the variables that makes all of the equations in the system true at the same time.

For a system with two variables, like $x$ and $y$, each linear equation represents a line on a graph. The solution to the system is the point where these lines intersect.

There are three possibilities for the solution to a system of linear equations:

1. One unique solution: The lines intersect at a single point. This is called a consistent and independent system.
2. No solution: The lines are parallel and never intersect. This is an inconsistent system.
3. Infinitely many solutions: The equations represent the same line. Every point on the line is a solution. This is a consistent and dependent system.

To build a solid intuition for these concepts, please read the introductory sections of the following resource from OpenStax.

While graphing is a great way to visualize what's happening, it's not always precise, especially if the solution involves fractions or complex decimals. For that, we need robust algebraic methods.

Solving Systems by Substitution

The substitution method is an intuitive algebraic approach. The core idea is to solve one equation for one variable and then substitute that expression into the other equation. This reduces a system of two equations with two variables down to a single equation with one variable, which we can then easily solve.

The following video provides a clear walkthrough of both substitution and elimination. We will focus on substitution first.

The general steps for the substitution method are:

1. Solve one of the equations for one of its variables (e.g., solve for $y$ in terms of $x$). Choose an equation where a variable has a coefficient of 1 or -1 to make this step easier.
2. Substitute the expression from step 1 into the other equation. This will give you an equation with only one variable.
3. Solve the resulting equation for that variable.
4. Substitute the value you found in step 3 back into one of the original equations (or the expression from step 1) to find the value of the other variable.
5. Verify your solution by plugging the values for both variables into both original equations to ensure they hold true.

This method can be extended to systems with more variables. For a system of three equations, you would solve for one variable, substitute it into the other two equations, and you'd be left with a familiar 2x2 system to solve.

Solving Systems by Elimination (or Addition)

The elimination method, sometimes called the addition method, works by manipulating the equations so that when you add them together, one of the variables is eliminated.

The key is to make the coefficients of one variable in two different equations exact opposites (e.g., $+3y$ and $-3y$). Often, this requires you to multiply one or both equations by a constant.

The following video is dedicated to the elimination method and is an excellent resource for mastering this technique, including its application to 3-variable systems.

The strategy for solving a system with three variables, say $x,y,z$, using elimination is a great example of breaking a complex problem into smaller, manageable parts:

1. Pair up two of the equations (e.g., the first and second) and use elimination to get rid of one variable (say, $y$). This leaves you with one new equation in terms of $x$ and $z$.
2. Pair up a different combination of equations (e.g., the first and third) and eliminate the same variable ($y$). This gives you a second new equation in terms of $x$ and $z$.
3. You now have a 2x2 system with your two new equations. Solve this system for $x$ and $z$ using either elimination or substitution.
4. Back-substitute the values of $x$ and $z$ that you just found into any of the original three equations to solve for $y$.

Handling Special Cases: Inconsistent and Dependent Systems

When you apply these algebraic methods, the results will clearly tell you which of the three solution types you are dealing with.

To summarize:

• If your process leads to a false statement (e.g., $9=13$), it is a contradiction. This means there is no solution, and the system is inconsistent.
• If your process leads to an identity (e.g., $0=0$), it means the equations are redundant. There are infinitely many solutions, and the system is dependent. You can express the solution set by writing one variable in terms of the other (e.g., $y=2x+5$, so the solution is $(x,2x+5)$).

Conclusion

In this lesson, we established a solid foundation for solving systems of linear equations. You learned to classify systems based on their number of solutions and how this relates to their graphical interpretation. Most importantly, you learned the step-by-step algebraic techniques of substitution and elimination, which allow for precise solutions to systems of any size.

Key Takeaways:

• A system of linear equations can have one unique solution, no solution, or infinitely many solutions.
• The substitution method involves isolating one variable and substituting its expression into another equation.
• The elimination method involves manipulating and adding equations to eliminate a variable.
• For systems with three or more variables, the strategy is to systematically eliminate variables to reduce the problem to a smaller, solvable system.
• Contradictions (like $0=5$) imply no solution, while identities (like $0=0$) imply infinite solutions.

In our next lesson, we will step into a different but equally fundamental area of mathematics: set theory. While it might seem like a change of pace, sets are the language used to formally define mathematical objects and concepts, including the solution sets you worked with today, the domains of functions, and the sample spaces used in probability. A solid grasp of set theory will provide the formal language needed for the advanced topics to come.