Systems of Equations
11 Problems • 4 sub-topics
Adalynn Le • 5/7/2026
Table of Contents
Introduction
Systems of equations are, by definition, a set of two or more equations. The "solutions" are the places where the graphs intersect, so the value for \(x\) yields the same value for the output \(f(x)\) or \(y\)
Linear Systems of Equations
We're going to start learning about systems of equations by talking about the simpelest type in order to learn the basic principles. A linear system of equations is set of two or more equations with a constant slope/rate of change. Unless the slope of two lines is the same, linear systems of equations will always meet at exactly one point.
A system of equations, regardless of the type, will always intersect at one point, or a set of points. Typically these points are not directly adjacent for any of the basic functions (linear, polynomial, exponential, radical, logarithmic, sinusodial), that is to say, they meet once and go separate ways before they ever intsersect again, if they do.
Solving Linear Systems of Equations
Most likely, you have learned how to solve linear systems of equations already. That being said, I believe that there is a lot to be learned about systems of equations in general that can be derived from just linear equations. There are \(3\) main ways for solving any system of equations:
Graphing
This technique is the easiest, arguably, because it does not require you to do any arithmetic. That being said, graphing is really only effective in certain situations because it requires accuracy and precision, it may require you to use tools (like a ruler or graphing paper), it is time consuming, and it only works when the solution is in a decent range. For example, you could graph the equation \(y=2x+1\) but not \(y=151x+1\) because the scale would need to be too large. For nonlinear systems of equations, it also becomes more unlikely to be able to use graphing beacuase the graphs are not straight and require you to draw a curve reliably and cleanly.
Substitution
Substitution is arguably the most straightforward form of algebraic manipulation and solving systems of equations. In this method, you would isolate one variable (similar to if you were solving for a constant value of \(x\) or \(y\)) but instead of finding it to be equivalent to a constant you express one variable as a function of another. Once you equate one variable to a expression, you can go to your ther equation and substitute in all values of that variable for that expression. In theory, this should reduce the number of variables by \(1\). As a general rule for solving via substitution, you need to have as many equations are variables. The only exception is in diophantine equations
Subtraction
The last example I want to bring up is one that only applies in some cases, but I find when applicable makes the situation a lot easier. It involves performing operations such as addition and subtraction with two or more whole equations with the hopes of eliminating a variable (or multiple). This means that the variables need to have the same coefficient and opposite signs. They do not nescessarily have to be written in this form, because you can also multiply by constants including \(-1\) to get to a form where you can subtract/add. This trick is especially helpful because it can sometimes allow you to solve nondiophantine equations where there are more variables than equations by eliminating multiple at once. Although not as striaghtforward and applicable as substitution, it is often a faster and more time-efficient solution when possible
The reason why I wanted to go over this now is that you will see these ideas and methods of solving featured in even more advanced and compelx systems of equations. Although the process in which you solve may be different, the way you solve will will always be one of the ways above. If you are not already experienced with using these for linear equations, I suggest to familiarize yourself with the process first before moving on to other functions
Make sure to remember that linear equations and systems equations represent a constant difference and a proportional relationship added to the \(y-\)intercept. This will come in handy later when we are interpreting equations.
Nonlinear Systems of Equations
The difference between linear and nonlinear equations really isn't that drastic. The methods for solving are the same, and solutions mean the same thing. There is only one significant difference that I would note about nonlinear equations: multiple solutions are possible, and the conditions for no solutions are more complex. Unfortunately, there is no clear way to find out how many solutions a system will have just by looking at it, but there are a few things that'll let you figure it out:
- Finding the minimum/maximum value of a graph. If the other line never goes near that area, or if in a close domain of values \(x\) it's output \(f(x)\) is very far off, they probably will not intersect
- If you are sure they intersect once/have already found one solution, you may not need to find the rest if you just need a number if you understand the shape and behavior of both graphs. For example, we know that a line will intersect a parabola twice UNLESS the intersection is at the vertex or unless there is no intersection. Thus, if you were solving for total
solutions, you wouldn't need to find all the solutions, just prove that there is one and it is not the vertex
Graph of \(x^2\)
Solutions
For systems of equations, there isn't really much to go about because most of the solving is intuitive and the knowledge comes from a basic understanding of functions and algebraic manipulation, since systems of equations are just combined functions with solutions
Solving Systems of Equations
This has alrady been gone over in the prior section on solving linear equations, but to summarize, every equation with a solution can be solved through graphing, substitution, or subtraction/addition. That being said, equations can and will have more or less than one solution. When equations have no solution, they will simplify down into an equation that is impossible. This typically means two constants that are not equal being described as equal to each other. When equations have multiple solutions, it must involve a polynomial, or the input \(x\) being raised to some power \(\neq 1\). By nature, a polynomial expression cannot hvae more solutions than the highest power that \(x\) is raised to. It can have less, but most will have as many solutions as the power of \(x\)
It was also mentioned that you can typically only find solutions in situations where there are as many variables as equations. That is true for most situations where all real numbers are considered, but we can perform cetain operations that limit the domain of solutions and thus make them easier or more possible to solve. One such example is a diophantine equation.
Diophantine Equations
Diophantine Equations are equations such that the solutions can only be integers. This may not provide the immediate solution to every equation, but it can severely limit equations with a domain of all real numbers, like in the example below.
A Standard Equation
Notice how there are seemingly infinite values for both of the variables, and that's considering we limited the range too!
A Diophantine Equation
y=
With this, there are only 10 total values for each variable, still dependent on each other. That makes an impossible equation possible
On the AMC 10, it isn't just a matter of solving equations; you are also expected to be able to interpret equations and use them best for your purposes. This involves being bale to know when and how to bind equations, as well as understanding equations in general. This is a skill that you can gain through working with modeling with word problems
Modeling Word Problems
Most of this information can also be found in our article about word problems but I want to use this to talk about modeling two or more equations when given a word problem and how to go about doing it.
When to use systems of equations
The whole point of the AMC 10 is to do math fast, which means you only want to use systems of equations when you need to. Only use a system of equation if you are following more than one variable, the question gives you different scenarios/terms, or if the question absolutely cannot be solved by logic/has an obvious answer. The most important thing to realize is that you only need a system of equations with multiple variables. If there is only one unknown you do not need it.
I would also like to remark that systems of equations are especially prevalent in geometry. Although they appear to be purely algebraic, in a lot of AMC 10 problems dealin with similarity and linked side lengths, they bear a great level of importance when solving for missing sides. If you ever know two values are the same or congruent, use a variable to represent it.
Writing the Right Equation
With word problems, it is important to understand what it is asking you to do. This involves understanding keywords and writing an equation that accurately represents the problem AND allows you to solve for the value you need. The best way to understand a word problem is to pick it apart into keywords. Keywords are words that indicate a word problem requires a certain operation. Here are a list of the most common keywords.
| Addition | Subtraction | Multiplication | Division |
|---|---|---|---|
| sum | difference | product | quotient |
| total | less | times | divided by |
| combined | decreased by | of | per |
| increased by | per unit |
Conclusion
Systems of equations are the most effective way to link and solve more than one variable. It goes hand in hand with algebraic manipulation and can be used to solve and model word problems. Although they may seem daunting, it is important to remember that systems of equations are just sets of two equations that overlap. The gap between simple linear equations and nonlinear equations is not far, and the methods for solving are very similar.
Question 1:
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