Solving systems of equations by graphing (even after Algebra 1)

academics algebra graphing mathematics systems of equations

When learning about lines in Algebra 1, you likely learned how to solve a system of (linear) equations, such as the following, by graphing:

Screen Shot 2023-07-23 at 4.20.12 PM

Recall that a point (x,y) is a solution to the system of equations only if plugging it in makes every equation in the system true. To solve by graphing, you graph both of the lines and then find the intersection point:

Screen Shot 2023-07-23 at 4.20.53 PM

Thus, the solution to the system of equations above is (3,2). [Don’t just take my word for it- verify it is a solution by plugging it into both equations!]

Finding the intersection on a graph may seem like just a quick trick to solve this one type of problem. But did you know that this method can continue to help you beyond Algebra 1? Here are a few ways how this concept comes up in Algebra 2, Pre-Calculus, and beyond.

Let’s say you are learning about linear vs. exponential growth in Algebra 2. Suppose you have money in two bank accounts represented by the following equations, with x being the time in months. How could you find out when they will contain the same amount of money?

Screen Shot 2023-07-23 at 4.21.24 PM

The bank account problem contains a more complicated system, but our goal is the same- we need to find the solutions! For each solution we find, y will give us how much money is in both accounts after x months. And the easiest way to find the solutions is to graph. If we graph both equations using technology, here is what we get:

Screen Shot 2023-07-23 at 4.21.57 PM

The graphs intersect at two points, so both accounts have the same amount of money at two different times! After 11.4 months, both accounts have $628.12. And after 54.8 months, both accounts contain $1495.83.

Note that linear systems typically only have a single solution, but other systems could have zero, one, two, three, or even up to an infinite number of solutions! No matter how many there are, you can find every (real) solution by graphing the equations and finding the intersections.

You can also use this graphing trick even when you only have a single equation (with a single variable). For example, let’s say you need to approximate all the values of x that make the following (ugly) equation true for your Pre-Calculus class:

Screen Shot 2023-07-23 at 4.22.34 PM

Just using algebra and brute force won’t work here since there is an “x” both inside and outside of the sine function- so we’ll need to graph! But how can we graph this without a “y=”? It turns out that we can rewrite this equation (of one variable) into a system of equations, where the left-hand side becomes the first equation and the right hand side becomes the second equation:

Screen Shot 2023-07-23 at 4.23.04 PM

Now, all we have to do is find the x-values of the solutions of this system. Since the y-values in each equation will be equal, that means that our original equation will be true as well! Graphing these two equations gives the following:

Screen Shot 2023-07-23 at 4.23.35 PM

Therefore, we can see that there are three solutions: approximately x=-3.768, x=0.896, and x=3.181.

Later, when you get to Calculus, you’ll learn how to find the area between two curves, and the points of intersection (solution) will help you find your lower and upper limits of integration. And in college, you might take a course in Linear Algebra, where you’ll upgrade from solving systems of two linear equations with two variables to solving systems of n linear equations with n variables (where n could be any whole number). It turns out that the solutions to these larger systems are quite important in the computational sciences!

So while you learn how to solve systems of equations by graphing in Algebra 1 using linear equations, don’t let this strategy gather dust afterwards! Keep it close at hand in your mathematical toolbox, and be ready to apply it to other types of equations as well!

James holds a PhD in Chemical Physics from Columbia and an MS in Chemistry from the University of Chicago. Previously, he graduated magna cum laude from Harvard College. He is currently a high school math teacher in the Boston area.


academics study skills MCAT medical school admissions SAT college admissions expository writing English MD/PhD admissions writing LSAT GMAT strategy GRE physics chemistry math biology graduate admissions ACT law school admissions interview prep test anxiety language learning academic advice MBA admissions premed personal statements homework help career advice AP exams creative writing MD study schedules Common Application summer activities test prep history computer science philosophy organic chemistry secondary applications supplements economics PSAT admissions coaching grammar mathematics research law statistics & probability psychology 1L ESL CARS SSAT covid-19 dental admissions legal studies logic games reading comprehension USMLE engineering Spanish calculus parents Latin verbal reasoning DAT PhD admissions case coaching excel mentorship political science AMCAS French Linguistics MBA coursework Tutoring Approaches academic integrity chinese medical school Anki DO English literature Social Advocacy admissions advice algebra astrophysics biochemistry business classics diversity statement genetics geometry kinematics mental health presentations quantitative reasoning skills study abroad time management work and activities IB exams ISEE MD/PhD programs STEM adjusting to college algorithms art history artificial intelligence athletics business skills careers cold emails data science functions gap year international students internships letters of recommendation logic mechanical engineering poetry resume science social sciences software engineering tech industry technical interviews trigonometry 2L 3L AAMC Academic Interest DMD EMT FlexMed Fourier Series Greek Health Professional Shortage Area Italian Lagrange multipliers London MD vs PhD MMI Montessori National Health Service Corps Pythagorean Theorem Python Sentence Correction Step 2 TMDSAS Taylor Series Zoom acids and bases amino acids analysis essay architecture argumentative writing art art and design schools art portfolios biomedicine brain teaser campus visits cantonese capacitors capital markets cell biology central limit theorem centrifugal force chemical engineering chess chromatography class participation climate change clinical experience community service constitutional law consulting cover letters curriculum dementia demonstrated interest dental school dimensional analysis distance learning electric engineering electricity and magnetism enrichment escape velocity european history executive function finance first generation student freewriting fun facts genomics graphing harmonics health policy history of medicine history of science hybrid vehicles hydrophobic effect ideal gas law induction infinite information sessions institutional actions integrated reasoning intern investing investment banking lab reports linear algebra linear maps mandarin chinese matrices mba medical physics meiosis microeconomics mitosis music music theory neurology neuroscience office hours operating systems organization pedagogy phrase structure rules plagiarism potential energy pre-dental proofs pseudocode psych/soc qualifying exams quantum mechanics relativity