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Introduction to Systems of Equations - Concept

Teacher/Instructor Alissa Fong
Alissa Fong

MA, Stanford University
Teaching in the San Francisco Bay Area


A system of equations is two or more equations that contain the same variables. A solutions to a system of equations are the point where the lines intersect. There are four methods to solving systems of equations: graphing, substitution, elimination and matrices. Solving systems of equations is an important concept that shows up first in Algebra I, but is built upon in upper-level math.

Solving systems of equations is a really really important skill in all of your Math classes. It shows up for the first in Algebra one but it continues in pretty much every course thereafter. So this is a really important idea that you're going to want to focus on in your first time through.
In Algebra class you'll probably only seen see two variables that are both raised to the first power but when you get into advanced Algebra or about Algebra two, you might start seeing things with x squared, y squared sometimes you have three or even four equations not just x and y but xyz and w or something like that but for Algebra most of the time we're just talking about two equations that have x and y in both of them.
The solution to a systems of equations, is the solution in both or all original equations. It's the point where the lines intersect. What that means is that if I take my x and y pair that I think is the solution and I substitute them into both original equations, both original equations will come out true they'll come out as equalities that's how I'll know if my work is correct. The other way I could check my work is by doing a graph and see if my solution point is where the lines cross. You'll get into that a lot once you start doing some practice problems. When you're asked to solve these, there's four methods that are commonly used.
The first method is to graph, you graph both lines see where they cross and that's your solution point. If you're not a good grapher don't worry there is other options for you. Substitution is where you get one variable isolated by itself and then substitute that expression into the other equation. That's what Substitution looks like and it's really commonly used when you have equations that are both in y=mx+b form.
A third method is called Elimination and that's where you look at the coefficients in front of x and y and you try to get coefficients in your two equations that are additive inverses. So like if my equation one has 3x+4y or something I want my equation two to have -3x, that way when I work with the equations together my positive 3x and -3x would be eliminated when I added them together. That's what an additive inverse is.
The fourth method is something that you don't see until you're more advanced in Math classes probably not until you're advanced Algebra or Algebra two class. So I'm going to cross that off we're not going to be going over that in Algebra one course but you will see that in your future.
So if you're a good grapher, this might be the one you choose, if you're good at using Algebra like you're pretty accurate when you write out all your Math homework and you don't lose negative signs you're pretty good with fractions and stuff like that Substitution or Elimination might be your favorite method. However, it's best to know how to do all the methods so that in case you get to your Math test and you have like a brain fart and you forget how to do Substitution, you're still able to use either Elimination or Graphing.

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